Optimal Control Problems with State Constraints in ... - Springer Link

Appl Math Optim 38:159–192 (1998)

© 1998 Springer-Verlag New York Inc.

Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion∗ H. O. Fattorini1 and S. S. Sritharan2 1 Department

of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

2 Code

754, Naval Command Control & Ocean Surveillance Center, San Diego, CA 92152-5000, USA

Abstract. Using nonlinear programming theory in Banach spaces we derive a version of Pontryagin’s maximum principle that can be applied to distributed parameter systems under control and state constrains. The results are applied to fluid mechanics and combustion problems. Key Words. Distributed parameter systems, Optimal controls, State constraints, Fluid flow control, Combustion control. AMS Classification.

1.

93E20, 93E25.

Introduction

The velocity y(t, x) = (y1 (t, x), . . . , ym (t, x)) and the pressure p(t, x) of a viscous fluid moving in a domain Ä ⊆ Rm with boundary 0 (m = 2, 3) are governed by the Navier−Stokes equations ∂y + (y · ∇)y = −∇ p(t, x) + ν1y(t, x) + u(t, x) ∂t ∇ ·y=0

(x ∈ Ä),

(x ∈ Ä),

(1.1) (1.2)

∗ This work was supported by the ONR Applied Analysis and Propulsion Programs under Grant 95WX20257.

160

H. O. Fattorini and S. S. Sritharan

y=0

(x ∈ 0),

y(0, x) = ζ (x)

(1.3) (x ∈ Ä),

(1.4)

where ν > 0 is the coefficient of kinematic viscosity, 1y = (1y1 , . . . , 1ym ) and u(t, x) denotes an external force, considered as a control. Distributed force control of this type can be realized in practice by electromagnetic forcing [28]–[30], [37]–[39]. Realistic constraints on the control are either pointwise, (0 ≤ t ≤ t¯, x ∈ Ä),

|u(t, x)| ≤ C

or integral, Z |u(t, x)| p d x ≤ C Ä

(1.5)

(0 ≤ t ≤ t¯ )

(1.6)

with 1 ≤ p < ∞, where | · | denotes a norm in Rm ; t¯ is the time at which the control process terminates. Both constraints can be written in the form u(t, · ) ∈ U

(0 ≤ t ≤ t¯ )

(1.7)

where U is a subset of (the space of m-vectors with coordinates in) L r (Ä), 1 ≤ r ≤ ∞. Natural state constraints are the velocity constraint (0 ≤ t ≤ t¯, x ∈ Ä),

|y(t, x)| ≤ C

(1.8)

the vorticity constraint |∇ × y(t, x)| ≤ C

(0 ≤ t ≤ t¯, x ∈ Ä),

(1.9)

and the stress (or, rather, deformation) constraint |∇y(t, x) + (∇y(t, x))T | ≤ C

(0 ≤ t ≤ t¯, x ∈ Ä);

(1.10)

in the latter | · | is an m × m matrix norm. All of the above are particular cases of the constraint Sy(t, x) ∈ M S ⊆ Rk

(0 ≤ t ≤ t¯, x ∈ Ä),

(1.11)

where S is a differential operator of the form Sy(x) =

m X

χ j (x)yj (x) +

j=1

m m X X

χ jk (x)

j=1 k=1

∂ yj (x) , ∂ xk

(1.12)

with χ j (x) and χ jk (x) k-vector functions defined in Ä. We also consider a separate state constraint at the terminal time t¯, given by T y(t¯, x) ∈ MY ⊆ Rl

(x ∈ Ä)

(1.13)

and called a target condition, with T a differential operator of the same form as S, T y(x) =

m X j=1

η j (x)yj (x) +

m m X X j=1 k=1

η jk (x)

∂ yj (x) , ∂ xk

(1.14)

Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion

161

η j (·) and η jk (·) l-vector functions defined in Ä. In certain situations (such as turbulence supression in selected regions) state constraints are only required in a subset Ä0 ⊆ Ä; this can be handled by multiplying the coefficients of S, T by the characteristic function of Ä0 . State conditions can also be of integral type, say Sy(t, ·) ∈ M S

(0 ≤ t ≤ t¯, x ∈ Ä)

(1.15)

where M S is a subset of (the space of k-vectors with coordinates in) L r (Ä) and the same applies to target conditions. The treatment is not limited to state constraints and target conditions expressed by linear differential operators; see Remark 6.3. Optimal problems for (1.1)–(1.4) deal with the minimization of a cost functional, in general of the form Z t y0 (t, u) = L(τ, y, u) dτ (1.16) 0

over a fixed or variable time interval 0 ≤ t ≤ t¯, where the control u(t, x) satisfies the control constraint and is such that the solution y(t, x) satisfies the state constraint; the kernels L(t, y, u) under consideration include, for instance, Z (1.17) L(t, y, u) = {|∇(y(t, x) − yd (x))|2 + |u(t, x)|2 } d x, Ä

which penalizes deviation from a desired state yd (x) and control cost. Optimal control problems with state and control constraints for evolution partial differential equations have been approached in two ways. The first way [15], [12]–[14], [16] is the one in this paper; it uses modeling of the equations as abstract differential equations in Banach spaces and formulates optimal problems as nonlinear programming problems in metric spaces. Pontryagin’s maximum principle appears as a Kuhn−Tucker theorem; see [13] for other references. The second approach is based on the theory of partial differential equations and covers as well steady state elliptic problems; see for instance [5], [6], and the references in [34]. This has also been used for the Navier−Stokes and related equations in [1], [6], [22], [23], [27], [43], and many other papers. Control problems for the Navier–Stokes equations are treated in this paper via modeling as abstract differential equations in L p spaces [41], [21], [24], [26], [46], [47]; the possibility of taking p arbitrarily large permits treatment of pointwise constraints on the velocity and on its derivatives via Sobolev embeddings (as far as we know, derivative constraints are not included in existing treatments). This point of view is closely related to that in [14]. However, the results in [14] are based on the theory of elliptic differential operators in spaces of continuous functions and do not apply to problems like (1.1)–(1.4) due to the fact that the projection operator into the space of divergence-free vectors, although bounded in L p spaces for 1 < p < ∞ is unbounded in spaces of continuous functions. For earlier works using the same approach in the “classical” case p = 2 and including control (but not state) constraints see [16]–[19], the second work [17] including a treatment of the Hamilton−Jacobi approach to feedback control; see also [44] for other references in this area. Without major modifications, the abstract model applies as well in other situations such as constant density combustion processes. In these, y(t, x) and p(t, x) denote

162


again velocity and pressure [4] and the functions ψ1 (t, x), . . . , ψn (t, x) are the scalar fields used in combustion; ψ1 (t, x) is the temperature field and ψ2 (t, x), . . . , ψn (t, x) are the components of the reactant and burnt product. Introducing the vector function ψ (t, x) = (ψ1 (t, x), . . . , ψn (t, x)) the equations are ∂y + (y · ∇)y = −∇ p(t, x) + ν1y(t, x) + u(t, x) ∂t ∇ ·y=0 y=0

(x ∈ Ä),

(x ∈ Ä),

(1.18) (1.19)

(x ∈ 0),

(1.20)

∂ψ ψ + (y · ∇)ψ ψ = f(ψ ψ ) + 10ψ (t, x) + v(t, x) ∂t

(x ∈ Ä),

(1.21)

where 10ψ = (ν1 1ψ1 , . . . , νn 1ψn ), v(t, x) is an additional n-vector control, and the components f j (ψ1 , . . . , ψn ) of f come from Arrhenius’ combustion law. The ψ j satisfy boundary conditions on 0, either Dirichlet as y, ψ =0

(x ∈ 0),

(1.22)

or of variational type. Control and state constraints, optimal problems and cost functionals are formulated in the same way as for the Navier–Stokes equations.

2.

The Navier–Stokes Equations as Abstract Differential Equations

We assume Ä is a bounded domain of class C (∞) in Rm , its boundary denoted by 0. The space L p (Ä)m consists of all m-vectors of functions in L p (Ä) and is equipped with its natural p-vector norm; the superindex m in other spaces indicates a similar vectorization. The space D(Ä)m consists of all m-vectors of test functions in Ä and j (Ä) ⊆ D(Ä)m is the subspace of all divergence free vectors (∇ · y = 0). Finally, X p (Ä)m is the closure of j (Ä) in L p (Ä)m , and G p (Ä)m ⊆ L p (Ä)m consists of all vectors ∇ p with p ∈ W 1, p (Ä). It was proved in [21] that L p (Ä)m = X p (Ä)m ⊕ G p (Ä)m ,

(2.1)

the direct sum orthogonal if p = 2; if p 6= 2 the sum is norm-direct, that is, the projection Pp : L p (Ä)m → X p (Ä)m is a bounded operator. The operator 1 p is the (m-vector) Laplacian in L p (Ä)m with domain D(1 p ) = 2, p W0 (Ä)m = {y ∈ W 2, p (Ä)m ; yj = 0 on 0}. The Stokes operator A p is defined by A p = ν Pp 1 p ,

D(A p ) = X p (Ä)m ∩ D(1 p ).

(2.2)

Applying the projection Pp to the Navier−Stokes system (1.1)–(1.4) with instantaneous values of the controls in L r (Ä) (r ≥ p) we obtain the abstract semilinear equation y0 (t) = A p y(t) + N (y(t)) + Bu(t),

y(0) = ζ ,

(2.3)

for the velocity y(t)(x) = y(t, x) in the space E = X (Ä) , where p

N (y) = −Pp (y · ∇)y,

B = Pp I p ,

m

(2.4)


163

and I p is the identity operator from L r (Ä) into L p (Ä). If 1 < p < ∞, the operator A p has a bounded inverse and generates a bounded, compact analytic semigroup Sp (·) in X p (Ä)m [24]–[26], [41], [42], [47]. We have [21] the following duality relations: (X p (Ä)m )∗ = X q (Ä)m ,

A∗p = Aq ,

Pp∗ = Pq ,

(2.5)

with 1/ p + 1/q = 1; in particular, A2 is self-adjoint in the Hilbert space X 2 (Ä)m . Since 0 belongs to the resolvent set of A p , fractional powers (−AR p )α can be defined; for Re α < 0 we may do so using the Dunford integral (2πi)−1 (−λ)−α R(λ; A) dλ over a suitable contour [48]. It is shown in [25] that the domains of the fractional powers (−A p )α can be exactly characterized by complex interpolation: we have D((−A p )α ) ≈ [X p (Ä)m , D(A p )]α . Also, D((−A p )α ) ≈ X p (Ä)m ∩ D((−1 p )α ) (here “≈” means equality of sets and equivalence of norms). We then use the fact that D((−1 p )α ) = [L p (Ä)m , D(1 p )]α (see [3] and references there) and the interpolation result ( 2α, p H0 (Ä)m (1/2 p < α < 1), 2, p p m m (2.6) [L (Ä) , W0 (Ä) ]α ≈ 2α, p m (Ä) (0 < α < 1/2 p). H For the definition of the spaces H s, p (Ä) see p. 250 of [45]; the space H0s (Ä) incorporates the boundary condition y = 0 on 0 for s > 1/ p. The interpolation equality (2.6) is (7) 2, p 2, p on p. 321 of [45] taking into consideration the fact that W0 (Ä)m ≈ H0 (Ä) [45, (11), (r ) p. 252]. The definition of the spaces C (Ä) for arbitrary real r is on p. 325 of [45] and the embedding (2.7) is (6) on p. 328 of [45]. Putting all of this together we obtain ( 2α, p X p (Ä)m ∩ H0 (Ä)m (1/2 p < α < 1), α D((−A p ) ) ≈ (2.7) p m 2α, p m (Ä) (0 < α < 1/2 p). X (Ä) ∩ H To deal with state constraints we use the embeddings (r ) H s, p (Ä) ⊂ → C (Ä),

(r ) W s, p (Ä) ⊂ → C (Ä)

(s − m/ p > r ).

(2.8)

The first embedding combined with (2.7) gives p m m D((−A p )α ) ⊂ → X (Ä) ∩ C0 (Ä) (1) p m m D((−A p )α ) ⊂ → X (Ä) ∩ C0 (Ä)

(m/2 p < α < 1), ( 12 + m/2 p < α < 1),

(2.9) (2.10)

the subindex in C0 and C0(1) indicating the boundary condition y = 0 on 0; note that the restrictions on α in both (2.9) and (2.10) imply those in the first line of (2.6) and those in (2.8) for r = 0, 1, respectively. Also, we use the fact that the spaces H s, p and W s, p are “comparable” in the sense that, if ε > 0, s, p s−ε, p H s+ε, p (Ä) ⊂ (Ä). → W (Ä) ⊂ →H

(2.11)

We study (2.3) through the abstract model y 0 (t) = Ay(t) + f (t, y(t)) + Bu(t),

y(0) = ζ,

(2.12)

164


where A is the infinitesimal generator of a bounded analytic semigroup S(t) in a reflexive separable Banach space E. The control space is F = X ∗ , X a separable Banach space, and we assume that B is a bounded operator with B: X ∗ → E,

B ∗ : E ∗ → X.

(2.13)

∗ Controls u(·) in (2.12) are elements of the space L ∞ w (0, T ; X ) of all X -weakly mea∗ surable a.e. bounded X -valued functions u(·). This space is equipped with the essential supremum norm, and is the dual of L 1 (0, T ; X ) with pairing hg(·), f (·)i = R hg(t), f (t)i dt. The control set U is a subset of X ∗ and the space of admissible controls ∗ Cad (0, T ; U ) consists of all u(·) ∈ L ∞ w (0, T ; X ) such that u(t) ∈ U a.e. in 0 ≤ t ≤ T. We assume that 0 ∈ ρ(A), so that the fractional powers (−A)α can be defined. These fractional powers are bounded for Re α < 0 and satisfy (−A)α = ((−A)−α )−1 (α ≥ 0) and (−A)α+β = (−A)α (−A)β for −∞ < α, β < ∞; moreover, S(t)E ⊆ D((−A)α ) and (−A)α S(t) is continuous in (E, E) for t > 0 ((E, F) is the space of all linear bounded operators from E into F equipped with the operator norm). Finally,

k(−A)α S(t)k ≤ Cα t −α e−ct

(t > 0, 0 ≤ α < 1).

(2.14)

If α ≥ 0 we define E α = D((−A)α ) equipped with the norm kykα = k(−A)α yk. Invertibility of (−A)α implies that E α is a Banach space. For α < 0, E α is the completion of E with respect to the norm k · kα . It follows from additivity of fractional powers that (−A)−ρ E α = E α+ρ

(2.15)

for ρ > 0 and α > 0, (−A)−ρ an isometric isomorphism. For α = −γ < 0, the operator (−A)−ρ is extended to E α = E −γ as follows. Let y ∈ E γ . Then there exists a sequence {yn } ∈ E such that kyn − yk−γ → 0. This implies that {(−A)−(γ −ρ) (−A)−ρ yn } = {(−A)−γ yn } is Cauchy in E, thus {(−A)−ρ yn } converges to z ∈ E −γ +ρ in the norm of E −γ +ρ and we define z = (−A)−ρ y. Obviously, we have k(−A)−ρ yk−γ +ρ = kyk−γ , so that (−A)−ρ is an isometric isomorphism and it can be shown [14] that (2.15) actually holds with α < 0. Given α, ρ ≥ 0 we say that f (t, y) satisfies Hypothesis Dα,ρ if f : [0, T ] × E α → E −ρ , the Fréchet derivative ∂ y f (t, y) ∈ (E α , E −ρ ) exists, and the function g: [0, T ] × E → E defined by g(t, η) = (−A)−ρ f (t, (−A)−α η)

(2.16)

satisfies: (a) g(t, η) is continuous in η for t fixed and strongly measurable in t for η fixed. (b) ∂η g(t, η)ζ = (−A)−ρ ∂ y g(t, (−A)−α η)(−A)−α is continuous in η for t, ζ fixed and strongly measurable in t for η, ζ fixed. (c) For every c > 0 there exist K (c), L(c) such that kg(t, η)k E ≤ K (c), (0 ≤ t ≤ T, kηk ≤ c).

k∂η g(t, η)k(E,E) ≤ L(c) (2.17)


165

Assuming that α + ρ < 1, the solution of (2.12) is defined by y(t) = (−A)−α η(t), where η(t) solves the integral equation Z t η(t) = (−A)α S(t)ζ + (−A)α+ρ S(t − τ )(−A)−ρ f (τ, (−A)−α η(τ )) dτ 0 Z t (2.18) + (−A)α S(t − τ )Bu(τ ) dτ. 0 ∗ ∗ On the last integral, note that if u(·) ∈ L ∞ w (0, T ; X ) and z ∈ E we have hz, Bg(t)i = ∗ ∗ hB z, g(t)i, so that Bu(·) is E -weakly measurable; since E is separable, Bu(·) is strongly measurable. Local solutions of (2.18) are constructed by successive approximations or as fixed points of a contraction map, the estimations based on (2.14), (2.17), and the generalized Gronwall’s lemma in [31]: if b ≥ 0, r > −1, R t and a(t), ϕ(t) are nonnegative and integrable in s ≤ t ≤ T with ϕ(t) ≤ a(t) + b s (t − τ )r ϕ(τ ) dτ , then there exists C Rt depending only on r, T such that ϕ(t) ≤ a(t) + Cb s (t − τ )r a(τ ) dτ. Solutions exist and are unique in the whole interval 0 ≤ t ≤ T or in a maximal interval of existence [0, tm ), tm < t at the end of which interval they blow up (that is, lim supt→tm kη(t)k = +∞). The treatment of the Navier–Stokes equations as abstract differential equations in X p (Ä)m follows [26]. The nonlinear term is

f (t, y) = N (y) = −Pp (y · ∇)y.

(2.19)

In particular, the result below is shown in [26]. Theorem 2.1. Let 1 < p < ∞, 1/ p+1/q = 1, α > 0, 0 ≤ ρ < 12 +m/2q, α +ρ > 12 , ρ + 2α > 12 + m/2 p. Then Pp (y · ∇)z maps (X p (Ä)m )α × (X p (Ä)m )α into (X p (Ä)m )−ρ and k(−A p )−ρ Pp (y · ∇)zk L p (Ä)m ≤ K k(−A p )α yk L p (Ä)m k(−A p )α zk L p (Ä)m .

(2.20)

We have Pp ((y + h)·∇)(y + h)− Pp (y·∇)y = Pp (y·∇)h+ Pp (h·∇)y+ Pp (h·∇)h, so that under the assumptions of Theorem 2.1 N (y) has a Fréchet derivative given by ∂N (y)h = −Pp (y · ∇)h − Pp (h · ∇)y

(2.21)

and y → ∂N (y) is continuous from (X p (Ä)m )α into ((X p (Ä)m )α , (X p (Ä)m )−ρ ); moreover, N and ∂N are locally bounded, so that N satisfies Hypothesis Dα,ρ . Some of the results below apply not only to (2.12) but to the more general model y 0 (t) = Ay(t) + f (t, y(t), u(t))

(2.22)

with f : [0, T ] × E α × U → E −ρ . This equation is equipped with an entirely arbitrary control set U and an admissible control space Cad (0, T ; U ) of U -valued functions. We assume that there exist α, ρ ≥ 0, α + ρ < 1 such that f (t, y) = f (t, y, u(t)) satisfies Hypothesis Dα,ρ for each u(·) ∈ Cad (0, T ; U ) with K (c) and L(c) independent of u(·).

166


Solutions of (2.22) are defined by means of the obvious extension of the integral equation (2.18). The space Cad (0, T ; U ) is equipped with the distance d(u(·), v(·)) = |{t; u(t) 6= v(t)}|,

(2.23)

where |e| denotes the Lebesgue outer measure of a (possibly nonmeasurable) set e. This distance makes Cad (0, T ; U ) a metric space. A measurable envelope [e] of a set e ⊆ [0, T ] is any measurable set [e] ⊇ e with |[e]| = |e|. We denote by y(t, u) the solution of (2.22) corresponding to the control u(·). ¯ exists in the Lemma 2.2. Let the control u(·) ¯ ∈ Cad (0, T ; U ) be such that y(t, u) interval 0 ≤ t ≤ t¯. Then there exists δ > 0 such that if the controls u(·), v(·) belong ¯ ≤ δ}, then y(t, u) and y(t, v) exist as to B(u, ¯ δ) = {u(·) ∈ Cad (0, T ; U ); d(u(·), u(·)) well in 0 ≤ t ≤ t¯ and Z (t − τ )−α−ρ dτ. (2.24) ky(t, v) − y(t, u)kα ≤ C [{τ ∈[0,t¯ ];u(τ )6=v(τ )}]

The proof uses the generalized Gronwall lemma and is similar to that of Lemma 5.1 in [9], where the assumptions on f (t, y, u) are actually weaker. All the assumptions for (2.22) are satisfied by (2.12) if we take as control set U a bounded subset of X ∗ and as Cad (0, T ; U ) the space of all X -weakly measurable U -valued functions.

3.

The Variational Equation and the Adjoint Variational Equation

We work under the assumptions in Section 2 on A and the space E. Since E is reflexive, A∗ is the infinitesimal generator of the strongly continuous semigroup S ∗ (t) = S(t)∗ and we define the spaces (E ∗ )α using the fractional powers (−A∗ )α = ((−A)α )∗ in the same way the E α are defined from the fractional powers (−A)α . The companion of (2.15) is (−A∗ )−ρ (E ∗ )α = (E ∗ )α+ρ

(3.1)

valid for −∞ < α < ∞, ρ ≥ 0. As a model for the variational equation we study y 0 (t) = (A + B(t))y(t) + f (t),

y(s) = ζ ∈ E −γ ,

(3.2)

and, as a model for the adjoint variational equation, z 0 (s) = −(A∗ + B(s)∗ )z(s) − g(s),

z(t¯ ) = z ∈ (E ∗ )−γ .

(3.3) ∗

In both equations, γ < 1. The model will be used for E = X (Ä) , E = X (Ä)m , A = A p , A∗ = Aq . We assume that there exist α, ρ ≥ 0 with α + ρ < 1 such that p

B(t) ∈ (E α , E −ρ ),

m

q

(3.4)

and that t → F(t)η = (−A)−ρ B(t)(−A)−α η is strongly measurable in E for η ∈ E; taking adjoints, this implies the corresponding properties for the operator function t → G(t)z = (−A∗ )−α B(t)∗ (−A∗ )−ρ z. Finally, we assume k(−A)−ρ B(t)(−A)−α k(E,E) ≤ K

(0 ≤ t ≤ T ),

(3.5)


167

which implies the corresponding bound for (−A∗ )−α B(t)∗ (−A∗ )−ρ ; in fact, we have (−A∗ )−α B(t)∗ (−A∗ )−ρ = ((−A)−α )∗ B(t)∗ ((−A)−ρ )∗ = (B(t)(−A)−α )∗ ((−A)−ρ )∗ = ((−A)−ρ B(t)(−A)−α )∗ after two applications of the adjoint rules on p. 301 of [40]. We do (3.2) assuming that f (t) ∈ E −δ with α +δ < 1, that (−A)−δ f (·) is a strongly measurable function with k(−A)−δ f (t)k E ≤ L

(0 ≤ t ≤ T )

(3.6)

and that α + γ < 1. The integral equation for η(t) = (−A)α y(t) is η(t) = (−A)α+γ S(t − s)(−A)−γ ζ Z t + (−A)α+ρ S(t − τ )(−A)−ρ B(τ )(−A)−α η(τ ) dτ s Z t + (−A)α+δ S(t − τ )(−A)−δ f (τ ) dτ.

(3.7)

s

The theory of (3.7) (and of all the following integral equations) R t is standard; they are solved by successive approximations using the beta formula s (t − τ )−α τ −β dτ = B(α, β)(t − s)1−α−β (see [31]). In particular, we have Theorem 3.1. Let ζ ∈ E −γ and f (·) ∈ L ∞ (0, T ; E −δ ). Then there exists a solution y(·) of (3.2) continuous in 0 < t ≤ T in the norm of E α and such that ky(t)kα ≤ C{(t − s)−α−γ kζ k−γ + k f (·)k L ∞ (0,T ;E−δ ) }

(0 < t ≤ T ).

If −γ = α, then y(·) is continuous in 0 ≤ t ≤ T and assumes the initial value ζ . The variation-of-constants formula for (3.2) is written using the solution Rα,γ (t, s) of the operator integral equation Rα,γ (t, s)ζ = (−A)α+γ S(t − s)ζ Z t + (−A)α+ρ S(t − τ )(−A)−ρ B(τ )(−A)−α Rα,γ (τ, s)ζ dτ.

(3.8)

s

As for all integral equations here, the solution behaves like its nonintegral term: the operator Rα,γ (t, s) is bounded and continuous in the norm of (E, E) for s < t and kRα,γ (t, s)k(E,E) ≤ C(t − s)−α−γ

(0 ≤ s < t ≤ t).

The solution η(t) of (3.7) is given by Z t Rα,δ (t, τ )(−A)−δ f (τ ) dτ. η(t) = Rα,γ (t, s)(−A)−γ ζ +

(3.9)

(3.10)

s

The proof of this formula follows from replacing η(t) (given by (3.10)) into the integral equation (3.7), and use the integral equation (3.8) for Rα,γ (t, s) and Rα,δ (t, s). We prove in the same way that if ε > 0 is such that α + γ + ε < 1, then Rα,γ (t, s)E ⊆ D((−A)ε ) for t > s and (−A)ε Rα,γ (t, s) is bounded and continuous in the norm of (E, E) with k(−A)ε Rα,γ (t, s)k(E,E) ≤ C(t − s)−α−γ −ε

(0 ≤ s < t ≤ T ).

(3.11)

168


That (−A)ε can be applied to the right side of (3.8) is obvious. The estimation is the same as that for (3.8) itself. Everything proved on (3.2) is good for (3.3), but we need an equation more general than (3.3), dz(s) = −{A∗ + B(s)∗ }z(s) ds − µ(ds),

z(t¯ ) = z ∈ (E ∗ )−γ

(3.12)

with γ < 1 and a measure forcing term µ(ds) taking values in the dual space (E α )∗ . This dual is identified by (E α )∗ = (E ∗ )−α

(3.13)

with pairing hz, yi(E ∗ )−α ×Eα = h(−A∗ )−α z, (−A)α yi E ∗ ×E .

(3.14)

The proof follows from (2.15) and (3.1); see also Lemma 4.4 of [14] for a more general result. The counterpart of the operator Rα,ρ (t, s) for the adjoint equation (3.3) is the operator ∗ (s, t) ∈ (E ∗ , E ∗ ) defined by the integral equation Rρ,α ∗ Rρ,α (s, t)z = (−A∗ )ρ+γ S(t − s)∗ z (3.15) Z t ∗ + (−A∗ )α+ρ S(σ − s)∗ (−A∗ )−α B(σ )∗ (−A∗ )−ρ Rρ,α (σ, t)z dσ s

and the variation-of-constants formula for υ(s) = (−A∗ )ρ z(s) is Z t ∗ ∗ −γ ∗ Rρ,δ (σ, t)(−A∗ )−δ g(σ ) dσ υ(s) = Rρ,γ (s, t)(−A ) z +

(3.16)

s

for g(·) taking values in (E ∗ )−δ . Standard manipulations with the integral equations show that ∗ (s, t) Rα,ρ (t, s)∗ = Rρ,α

(0 ≤ s < t ≤ t¯ ).

(3.17) ∗

With this motivation in mind, the solution of (3.12) with z ∈ (E )−γ is defined as z(s) = (−A∗ )−ρ υ(s), where Z t ∗ ∗ −γ ∗ Rρ,α (σ, t)(−A∗ )−α µ(dσ ) υ(s) = Rρ,γ (s, t)(−A ) z + = υh (s) + υi (s).

s

(3.18) ∗

The forcing measure µ(ds) belongs to the space 6(0, T ; (E )−α ) of all countably additive bounded (E ∗ )−α -valued measures defined in the field generated by the closed subsets of [0, T ], this space equipped with the total variation norm (see [8], [12], and [14]). The space 6(0, TR ; (E ∗ )−α ) is isomorphic to the dual C(0, T ; E α )∗ under the duality map hy, µic,α = hy(t), µ(dt)iα , h · , · iα the duality of E α and (E α )∗ = (E ∗ )−α . For each s, the integral on the right-hand side of (3.18) is understood as follows: υi (s) is the unique element of E ∗ with Z t¯ hy, υi (s)i = hRα,ρ (σ, s)y, ν(dσ )i (y ∈ E) (3.19) s

for ν = (−A∗ )−α µ ∈ 6(0, t¯; E ∗ ). The two following results are particular cases of Lemmas 4.1 and 4.2 in [14]:


169

Theorem 3.2. Let z ∈ (E ∗ )−γ and µ ∈ 6(0, t¯; (E ∗ )−α ). Then υi (s) exists, belongs to E ∗ a.e. in 0 ≤ s ≤ t¯, is strongly measurable, and satisfies Ã ! Z t¯ (σ − s)−α−ρ kµk(E ∗ )−α (dσ ) kυi (s)k E ∗ ≤ C (t¯ − s)−ρ−γ kzk−γ + s

= C((t¯ − s)

−ρ−γ

kzk−γ + ω(s)),

(3.20)

where kµ(dσ )k(E ∗ )−α is the total variation of µ(dσ ). The function ω(s) on the right side of (3.20) belongs to L 1 (0, t¯ ). Let g(·) ∈ L ∞ (0, T ; E), ν ∈ 6(0, t¯, E ∗ ). Then

Lemma 3.3. Z t¯ ¿Z 0

t

À Rα,ρ (t, s)g(s) ds, ν(dt) =

0

Z t¯ *

Z

t¯

g(s), 0

+ ∗

Rα,ρ (t, s) ν(dt) ds.

s

In the results above, (3.2) stands for the variational equation of (2.22): ξ 0 (t) = {A + ∂ y f (t, y(t, u), u(t))}ξ(t) + f (t),

(3.21)

where y(t, u) is required to exist in 0 ≤ t ≤ t¯. We use the notation Rα,ρ (t, s; u) to indicate dependence of the equation on u. Lemma 3.4. Let u(·) ∈ Cad (0, t¯; U ) be such that y(t, u) exists in 0 ≤ t ≤ t¯. Then the operator Rα,ρ (t, s; u) depends continuously on t, s, u in the norm of (E, E) in the set {(t, s); 0 ≤ s < t ≤ t¯ } × B(u, ¯ δ), where B(u, ¯ δ) is the ball in Lemma 2.2. See [9] or [14] for a proof.

4.

Nonlinear Programming. Patch Perturbations

The minimum principle is obtained using the theory of the abstract nonlinear programming problem minimize f 0 (u) subject to

(4.1)

f (u) ∈ Y = target set

(4.2)

[20], [9], where f : V → E, f 0 : V → R (V is a complete metric space, E is a Banach space), Y ⊆ E. All results needed are in [20]; the versions used here [9] relax the assumptions on f , f 0 in [20] so that they can be applied to the problems in this paper. The time optimal problem is handled via the abstract time optimal problem, where {Vn } is a sequence of metric spaces, E is a Banach space, Y ⊆ E, f n : V → E such that f n (Vn ) ∩ Y = ∅,

(4.3)

and we try to characterize the sequences {u¯ n }, u¯ n ∈ Vn such that dist( f n (u¯ n ), Y ) → 0

as n → ∞.

(4.4)

170


A sequence {u¯ n }, u¯ n ∈ Vn that satisfies (4.4) is called an optimal sequence. We assume all functions f, f n , are everywhere defined and continuous, and the target set Y is closed. Let g: V = metric space → E = normed space. We say that ξ ∈ E is a (one sided) directional derivative of g at u ∈ V if there exists a function u: [0, δ] → V (δ > 0) with d(u(h), u) ≤ h and g(u(h)) = g(u) + hξ + o(h)

as

h →0+.

The set of all directional derivatives of g at u is called Der g(u). Theorem 4.1 below is a result of Kuhn–Tucker type for (approximate) solutions of (4.1)–(4.2). We call m the minimum of f 0 (u) subject to the target condition (4.2) and assume that −∞ < m < ∞.

(4.5)

An approximate or suboptimal solution of (4.1)–(4.2) is a sequence {u¯ n } ⊆ V such that lim sup f (u¯ n ) ≤ m, n→∞

lim dist( f (u¯ n ), Y ) = 0.

n→∞

(4.6)

A sequence { y¯ n } ∈ Y is associated with {u¯ n } if k f (u¯ n ) − y¯ n k → 0; it follows that there exists a sequence {εn } of positive numbers with εn → 0 and such that f (u¯ n ) ≤ m + εn ,

k f (u¯ n ) − y¯ n k ≤ εn .

(4.7)

We write f(u) = ( f 0 , f )(u) = ( f 0 (u), f (u)) ∈ E = R × E. We indicate by K Y ( y¯ ) (resp. TY ( y¯ )) the contingent (resp. tangent) cone to Y at y¯ . The tangent cone is defined as follows: w ∈ TY ( y¯ ) if, for every sequence { y¯k } ⊆ Y with y¯k → y¯ and every sequence {h k } ⊆ R+ with h k → 0, there exists a sequence {yk } ⊆ Y with yk − y¯k →w hk

as k → ∞.

(4.8)

On the other hand, the contingent cone K Y ( y¯ ) consists of all w such that (4.8) holds for y¯k = y¯ and some sequences {yk } ∈ Y, {h k } ∈ R+ with yk → y¯ , h k → 0. Finally, lim infn→∞ Z n ({Z n } is a sequence of sets in E) is the set of all limits z = limn→∞ z n with z n ∈ Z n . If Z ⊆ E, then the cone Z − ⊆ E ∗ normal to Z consists of all y ∗ ∈ E ∗ with hy ∗ , yi ≤ 0 (y ∈ Z ). Theorem 4.1. Let {u¯ n } ⊆ V be a suboptimal solution of (4.1)–(4.2), and let { y¯ n } ⊆ Y be a sequence associated with {u¯ n }. Then there exist sequences {u˜ n } ⊆ V, { y˜ n } ⊆ Y with √ (4.9) d(u˜ n , u¯ n ) + k y˜ n − y¯ n k ≤ 2 εn and such that, for every sequence {Dn } of convex sets such that Dn ⊆ Der f(u˜ n ) = Der( f 0 , f )(u˜ n ) and every ρ > 0 there exists a sequence {z 0n } ∈ R and a sequence {z n } ∈ E ∗ with √ 2 + kz n k2 = 1, z 0n ≥ 0, z 0n ξ0n + hz n , ξ n − w n i ≥ −2 εn (1 + ρ) (4.10) z 0n for (ξ0n , ξ n ) ∈ Dn , w n ∈ TY ( y˜ n ) ∩ B(0, ρ). If (z 0 , z) is an (R × E)-weak limit point of


{(z 0n , z n )} in R × E ∗ , then ³ ´− z ∈ lim inf TY ( y˜ n ) , z 0 ≥ 0, n→∞

z 0 ξ0 + hz, ξ i ≥ 0,

171

(4.11) (4.12)

for (ξ0 , ξ ) ∈ lim infn→∞ Dn . Let {u¯ n } be an optimal sequence for the abstract time optimal problem. A sequence { y¯ } ∈ Y is associated with {u¯ n } if n

k f n (u¯ n ) − y¯ n k = εn → 0

as n → ∞.

(4.13)

Theorem 4.2. Let {u¯ n }, u¯ n ∈ Vn be an optimal sequence, and let { y¯ n } ⊆ Y be a sequence associated with {u¯ n }. Then there exist sequences {u˜ n }, u˜ n ∈ Vn and { y˜ n } ⊆ Y such that √ dn (u˜ n , u¯ n ) + k y˜ n − y¯ n k ≤ εn (4.14) (dn is the distance in Vn ) and such that, for every sequence of convex sets {Dn }, Dn ⊆ Der f n (u˜ n ) and every ρ > 0 there exists a sequence {z n } ∈ E ∗ such that √ kz n k = 1, hz n , ξ n − w n i ≥ − εn (1 + ρ) (4.15) for ξ n ∈ Dn , wn ∈ TY ( y˜ n ) ∩ B(0, ρ). If z is an E-weak limit point of {z n }, then we have ´− ³ (4.16) z ∈ lim inf TY ( y˜ n ) n→∞

and hz, ξ i ≥ 0

(4.17)

for ξ ∈ lim infn→∞ Dn . For proofs of these results see Theorems 2.8 and 2.4 of [9]. They are useful when one can ensure that the multipliers (z 0 , z) and z in the Kuhn–Tucker inequalities (4.12) and (4.17) are nonzero, a question addressed below. A sequence {Q n } of subsets of a Banach space is precompact if every sequence {qn }, qn ∈ Q n has a convergent subsequence. Lemma 4.3. (a) Let {Dn } and ρ be as in Theorem 4.2, and let {Q n } be a precompact sequence such that the set ∞ \ {Dn − TY ( y˜ n ) ∩ B(0, ρ) + Q n } (4.18) 1= n=n 0

contains an interior point for n 0 large enough. Then the sequence {z n } in Theorem 4.2 corresponding to {Dn } and ρ cannot have zero E-weak limit points, so that the multiplier z in (4.17) is not zero.

172


(b) Let {Dn } and ρ be as in Theorem 4.1, Dn = 5(Dn ) (5 = projection into the second coordinate), and assume that (4.18) contains an interior point for n 0 large enough. Then the sequence (z 0n , z n ) in Theorem 4.1 corresponding to {Dn } and ρ cannot have zero weak limit points, so that the multiplier (z 0 , z) in (4.12) is nonzero. For a proof see Corollaries 2.6 and 2.9 of [9]. All of this will be applied to the functions f(u) = (y(·, u), y(t¯, u)),

f 0 = y0 (t¯, u),

(4.19)

with V = Cad (0, t¯; U ), E = C(0, t¯; E α ) × E α , where y(t, u) is the solution of (2.22) corresponding to u(·) and y0 (t, u) is a cost functional to be specified later. A sequence u(·) ⊆ Cad (0, t¯; U ) is stationary if there exists a set e with |e| = 0 such that for every t ∈ [0, T ]\e there exists n (depending on t) such that u n (t) = u n+1 (t) = u n+2 (t) = · · ·. The space Cad (0, t¯; U ) is saturated if the pointwise limit of every stationary sequence belongs to Cad (0, t¯; U ). It can be shown [9] that saturation implies completeness in the distance (2.22). Directional derivatives of f and of f n will be constructed by means of patch perturbations (introduced in [33]) and defined as follows. Let u(·) ∈ Cad (0, T ; U ), let v(·) = (v1 (·), . . . , vm (·)) be an arbitrary collection of m elements of Cad (0, T ; U ), and let e = (e1 , . . . , em ) be a collection of m pairwise disjoint sets in [0, T ] (m is arbitrary). The patch perturbation u e,v of u(·) corresponding to e, v is u e,v (t) = v j (t) (t ∈ e j , j = 1, . . . , m), u e,v (t) = u(t) elsewhere. If d is the distance (2.23) we have ¯[ ¯ ¯ ¯ d(u e,v , u) ≤ ¯ e j ¯ . (4.20) The admissible control space Cad (0, T ; U ) is patch complete if every patch perturbation of a control u(·) belongs to Cad (0, T ; U ). Lemma 2.2 guarantees S that if y(t, u) exists in 0 ≤ t ≤ t¯, then y(t, u e,v ) exists as well in 0 ≤ t ≤ t¯ for | e j | sufficiently small. Computation of directional derivatives is based on the following result, and requires Hypothesis Dα,ρ in Section 2 with α + ρ < 1 for f (t, y) = f (t, y, u(t)), with K (c) and L(c) in (2.17) independent of u(·) ∈ Cad (0, T ; U ). Theorem 4.4. Let u(·) ¯ ∈ Cad (0, T ; U ) be such that y(t, u) exists in 0 ≤ t ≤ t¯, let p = ( p1 , . . . , pm ) be a probability vector, and let v(·) = (v1 (·), . . . , vm (·)) be a collection of elements of Cad (0, T ; U ). Then, S for each h small enough, there exists e(h) = (e1 (h), . . . , em (h)) with |e(h)| = | e j (h)| ≤ h t¯ such that y(t, u e,v ) exists in 0 ≤ t ≤ t¯ for sufficiently small h and 1 ξ(t, u, ¯ p, v) = lim (y(t, u¯ e(h),v ) − y(t, u)) ¯ (4.21) h→0+ h exists uniformly in 0 ≤ t ≤ t¯ in the norm of E α , where ξ(t) = ξ(t, u, ¯ p, v) solves the variational equation ¯ u(t))}ξ(t) ¯ ξ 0 (t) = {A + ∂ y f (t, y(t, u), m X + p j { f (t, y(t, u), ¯ v j (t)) − f (t, y(t, u), ¯ u(t))}, ¯ j=1

ξ(0) = 0. (4.22)


173

The proof is the same as that of Theorem 5.1 of [14] and is omitted. ¯ p, v) is The variation-of-constants formula for υ(t, u, ¯ p, v) = (−A)α ξ(t, u, υ(t, u, ¯ p, v) Z m X = pj

t

Rα,ρ (t, τ, u) ¯

0

j=1

× (−A)−ρ { f (τ, (−A)−α η(τ, u), ¯ v j (τ )) − f (τ, (−A)−α η(τ, u), ¯ u(τ ¯ ))} dτ. (4.23)

5.

The Minimum Principle with State Constraints

We prove it for (2.22), where f (t, y, u(t)) satisfies the assumptions in Section 2. The state constraint and target condition are y(t) ∈ M ⊆ E α

(0 ≤ t ≤ t¯ ),

y(t¯ ) ∈ Y ⊆ E α ,

with Y and M closed in E α , and the cost functional Z t y0 (t, u) = f 0 (τ, y(τ, u), u(τ )) dτ

(5.1)

(5.2)

0

with f 0 : [0, T ]× E α ×U → R. We say that f 0 (t, y) = f 0 (t, y, u(t)) satisfies Hypothesis Dα0 if the Fréchet derivative ∂ y f 0 (t, y) ∈ (E α )∗ = (E ∗ )−α exists and (a) f 0 (t, y) is measurable in t for y fixed and continuous in y for t fixed, (b) ∂ y f 0 (t, y)z is continuous in y for t, z fixed and strongly measurable in t for y, z fixed, (c) for every c > 0 there exists K 0 (c), L 0 (c) such that | f 0 (t, y)| ≤ K 0 (c), k∂ y f 0 (t, y)k(E ∗ )−α ≤ L 0 (c) (0 ≤ t ≤ T, kykα ≤ c),

(5.3)

and we assume below that K 0 (c), L 0 (c) are independent of u ∈ Cad (0, T ; U ). We note that these conditions are trivially satisfied by the functional f 0 (t, y, u) = L(t, y, u) given by (1.17). In fact, taking α = 12 and using the first relation (2.7) we see that Z ∂y L(t, y, u)h = 2 {∇(y(x) − yd (x)) · ∇h(x)} d x (5.4) Ä

has all the required properties. The first result is a complement to Theorem 4.4. ¯ exists in 0 ≤ t ≤ t¯. Then Theorem 5.1. Let u(·) ∈ Cad (0, t¯; U ) be such that y(t, u) the sets e j (h) in Theorem 4.4 can be chosen in such a way that the limit 1 (y0 (t, u¯ e(h),v ) − y0 (t, u)) ¯ h→0+ h

ξ0 (t, u, ¯ p, v) = lim

(5.5)

174


exists uniformly in 0 ≤ t ≤ t¯, where Z t ξ0 (t, u, ¯ p, v) = h∂ y f 0 (τ, y(τ, u), ¯ u(τ ¯ )), ξ(τ, u, ¯ p, v)iα dτ 0

+ Z = 0

+

m X

Z

t

pj

{ f 0 (τ, y(τ, u), ¯ v j (τ )) − f 0 (τ, y(τ, u), ¯ u(τ ¯ ))} dτ

0

j=1 t

h(−A∗ )−α ∂ y f 0 (τ, y(τ, u), ¯ u(τ ¯ )), υ(τ, u, ¯ p, v)i dτ

m X

Z

t

{ f 0 (τ, y(τ, u), ¯ v j (τ ))− f 0 (τ, y(τ, u), ¯ u(τ ¯ ))} dτ. (5.6)

pj

j=1

0

The proof is straightforward. In this formula h · , · iα indicates the duality of E α and ¯ p, v) = (−A)α ξ(τ, u, ¯ p, v). (E ∗ )−α and υ(τ, u, Below, the control space Cad (0, T ; U ) is assumed saturated, so that it is complete with respect to the metric (2.23). We also assume Cad (0, T ; U ) is patch complete (Section 4) and define a set M(t¯ ) ⊆ C(0, t¯; E α ) by M(t¯ ) = {y(·) ∈ C(0, t¯; E α ); y(t) ∈ M (0 ≤ t < t¯ )}. Theorem 5.2 (the Minimum Principle). Let u(·) ¯ ∈ Cad (0, T ; U ) be an optimal control. Then there exists (z 0 , µ, z) ∈ R × 6(0, T ; (E ∗ )−α ) × (E ∗ )−α with ³ ´− µ ∈ lim inf TM(t¯ ) ( y˜ n (·)) ,

z 0 ≥ 0,

n→∞

³ ´− z ∈ lim inf TY ( y˜ n ) , n→∞

(5.7)

¯ where { y˜ n (·)} (resp. { y˜ n }) is a sequence in M(t¯ ) (resp. in Y ) such that y˜ n (·) → y(·, u) ¯ in E α ) and such that if z¯ (s) is the solution of in C(0, t¯; E α ) (resp. y˜ n → y(t¯, u) ∗ d z¯ (s) = −{A∗ + ∂ y f (s, y(s, u), ¯ u(s)) ¯ }¯z (s) ds ¯ u(s)) ¯ ds − µ(ds), − z 0 ∂ y f 0 (s, y(s, u),

z(t¯ ) = z,

(5.8)

in 0 ≤ t ≤ t¯, then Z

t¯

z0

{ f 0 (σ, y(σ, u), ¯ v(σ )) − f 0 (σ, y(σ, u), ¯ u(σ ¯ ))} dσ

0

Z

+

t¯

h¯z (s), f (σ, y(σ, u), ¯ v(σ )) − f 0 (σ, y(σ, u), ¯ u(σ ¯ ))iρ dσ ≥ 0.

(5.9)

0

Proof. We apply the nonlinear programming theory in Section 4 to the functions (4.19). The metric space V is the ball B(u, ¯ δ) in Lemma 2.2, where f is continuous; continuity ¯ δ) of f0 is shown in a similar way. Theorem 4.1 produces the sequences {u˜ n (·)} ∈ B(u, and {( y˜ n (·), y˜ n )} ⊆ M(t¯ ) × Y in the statement. We use as Dn the set of all elements of R × C(0, t¯; E α ) × E α of the form (t¯−1 ξ0 (t¯, u˜ n , p, v), t¯−1 ξ(·, u˜ n , p, v), t¯−1 ξ(t¯, u˜ n , p, v))

(5.10)


175

for p, v arbitrary. With v fixed, the limit of (5.10) in R × C(0, t¯; E α ) × E α is (t¯−1 ξ0 (t¯, u, ¯ p, v), t¯−1 ξ(·, u, ¯ p, v), t¯−1 ξ(t¯, u, ¯ p, v)).

(5.11)

This follows from the variation-of constants formula (4.23) and continuity of the solution operator (Lemma 3.4). Once ρ and {Dn } have been selected, Theorem 4.1 gives sequences {z 0n } ⊆ R, {µn } ⊆ 6(0, t¯; (E ∗ )−α ), and {z n } ⊆ (E ∗ )−α such that 2 z 0n ≥ 0, z 0n + kµn k26(0,T ;(E ∗ )−α ) + kz n k2(E ∗ )−α = 1, n z 0n ξ0 (t¯, u˜ , p, v) + hµn , ξ(·, u˜ n , p, v)ic,α + hz n , ξ(t¯, u˜ n , p, v)iα ≥ −δn ,

(5.12)

δn → 0; the first angled bracket is the duality of C(0, t¯; E α ) and 6(0, t¯; (E ∗ )−α ). If (z 0 , µ, z) is a (R × C(0, t¯; E α ) × E α )∗ -weak limit point of (z 0n , µn , z n ), then we have z 0 ξ0 (t¯, u, ¯ p, v) + hµ, ξ(·, u, ¯ p, v)ic,α + hz, ξ(t¯, u, ¯ p, v)iα ≥ 0.

(5.13)

For single patches (m = 1, p = {1}, v(·) = {v(·)}) we obtain z 0 ξ0 (t¯, u, ¯ v) + hµ, ξ(·, u, ¯ v)ic,α + hz, ξ(t¯, u, ¯ v)iα ≥ 0

(5.14)

using the shorthand ξ(t, u, ¯ v) = ξ(t, u, ¯ p, v). Taking advantage of (5.5), Z

t¯

z0

{ f 0 (σ, y(σ, u), ¯ v(σ )) − f 0 (σ, y(σ, u), ¯ u)} ¯ dσ

0

Z t¯ ¿ ¯ u(τ ¯ )), (−A)−α ∂ y f 0 (τ, y(τ, u), + z0 0 À Z τ −ρ Rα,ρ (τ, σ ; u)(−A) ¯ { f (σ, y(σ, u), ¯ v(σ )) − f (σ, y(σ, u), ¯ u(σ ¯ ))} dσ dτ 0

+

Z t¯ ¿

(−A∗ )−α µ(dt),

0

Z

t

−ρ

Rα,ρ (t, σ ; u)(−A) ¯

À { f (σ, y(σ, u), ¯ v(σ )) − f (σ, y(σ, u), ¯ u(σ ¯ ))} dσ

0

¿

+ (−A∗ )−α z, Z

t¯

−ρ Rα,ρ (t¯, σ ; u)(−A) ¯ { f (σ, y(σ, u), ¯ v(σ )) − f (σ, y(σ, u), ¯ u(σ ¯ ))} dσ

À

0

≥ 0.

(5.15)

After a switch in the order of integration (justified by Lemma 3.3 in the fourth integral) we obtain (5.9). This ends the proof. If e is the set of all Lebesgue points of all functions h¯z (s), f (s, y(s, u), ¯ u(s))i, ¯ ¯ u(s)), ¯ h¯z (s), f (s, y(s, u), ¯ v)i, f 0 (s, y(s, u), ¯ v) (v ∈ U ) we obtain the f 0 (s, y(s, u),

176


pointwise minimum principle z 0 f 0 (s, y(s, u), ¯ u(s)) ¯ + h¯z (s), f (s, y(s, u), ¯ u(s))i ¯ ρ = min{z 0 f 0 (s, y(s, u), ¯ v) + h¯z (s), f (s, y(s, u), ¯ v)iρ }

(5.16)

v∈U

for s ∈ e plugging spike perturbations u s,h (σ ) in (5.9) (u s,h (σ ) = v (s − h ≤ σ ≤ s), ¯ ) elsewhere) and letting h → 0. u s,h (σ ) = u(σ For the time optimal problem we use Theorem 4.2 with Vn = Cad (0, tn ; U ), {tn } a sequence in [0, t¯ ) with tn → t¯, and fn (u) = (y(·, u), y(t¯, u)) ∈ C(0, tn ; E α ) × E α . We obtain sequences {µn } ⊆ 6(0, T ; (E ∗ )−α ) and {z n } ⊆ (E ∗ )−α such that kµn k26(0,T ;(E ∗ )−α ) + kz n k2(E ∗ )−α = 1,

(5.17)

hµn , ξ(·, u˜ n , p, v)ic,α + hz n , ξ(t¯, u˜ n , p, v)iα ≥ −δn , with δn → 0. If (µ, z) is a (C(0, t¯, E α ) × E α )-weak limit point of (µn , z n ), then hµ, ξ(·, u, ¯ p, v)ic,α + hz, ξ(t¯, u, ¯ p, v)iα ≥ 0,

(5.18)

and operating in the same way as after (5.15) we obtain Z

t¯

h¯z (s), f (σ, y(σ, u), ¯ v(σ )) − f 0 (σ, y(σ, u), ¯ u(σ ¯ ))iρ dσ ≥ 0

(5.19)

0

for all v(·) ∈ Cad (0, t¯; U ), where ∗ d z¯ (s) = −{A∗ + ∂ y f (s, y(s, u), ¯ u(s)) ¯ }¯z (s) ds − µ(ds),

z¯ (t¯ ) = z.

(5.20)

As above, we obtain the pointwise minimum principle h¯z (s), f (s, y(s, u), ¯ u(s))i ¯ z (s), f (s, y(s, u), ¯ v)iρ ρ = minh¯ v∈U

(5.21)

in the set e of Lebesgue points of all functions h¯z (s), f (s, y(s, u), ¯ u(s))i ¯ ρ and h¯z (s), f (s, y(s, u), ¯ v)iρ (v ∈ U ). We look below at conditions that prevent the multipliers (z 0 , µ, z) and (µ, z) from ˜ exists in 0 ≤ t ≤ t˜. We denote by being zero. Let u˜ ∈ Cad (0, t˜; U ) be such that y(t, u) 4(0, t˜; U, u) ˜ ⊆ C(0, t˜; E α ) the set of all trajectories of the inhomogeneous variational equation ξ 0 (t) = A(t) + ∂ y f (t, y(t, u), ˜ u(t))ξ(t) ˜ + { f (t, y(t, u), ˜ u(t)) − f (t, y(t, u), ˜ u(t))}, ˜

ξ(0) = 0;

(5.22)

where u(·) ∈ Cad (0, t˜; U ). By formula (4.23), conv 4(0, t˜; U, u) ˜ coincides with the set {ξ(·, u, ˜ p, v)} (all probability vectors p and all vectors v of elements of Cad (0, t˜; U )). The set of values of these trajectories at time t˜ is the reachable space R(0, t˜; U, u) ˜ ⊆ Eα . Finally, we denote by 4(0, t˜; U, u) ˜ ⊆ C(0, t˜; E α ) × E α the set of all (ξ(·), ξ(t˜ )), ξ(·) ∈ 4(0, t˜; U, u). ˜ The following result is a consequence of Lemma 4.3.


177

Lemma 5.3. Assume that for every sequence {u˜ n (·)} ⊆ Cad (0, t¯; U ) with u˜ n (·) → u(·) ¯ ¯ y(t¯, u)) ¯ there and every sequence {( y˜ n (·), y˜ n )} ⊆ M(t¯ )×Y with ( y˜ n (·), y˜ n ) → (y(·, u), exists ρ > 0 and a precompact sequence {Qn }, Qn ⊆ C(0, t¯; E α ) × E α , such that ∞ \

{1n + Qn }

(5.23)

n=n 0

contains an interior point in C(0, T ; E α ) × E α for n 0 large enough, where 1n = t¯−1 conv 4(0, t¯; U, u˜ n ) − (TM(t¯ ) ( y˜ n (·)) × TY ( y˜ n )) ∩ B(0, ρ),

(5.24)

B(0, ρ) is the ball of center 0 and radius ρ in C(0, t¯; E α ) × E α . Then any weak limit point (z 0 , µ, z) of the sequence (z 0 , µn , z n ) is nonzero. The result for the time optimal problem is the same, with t¯ replaced by a sequence {tn } ⊆ [0, t¯ ) with tn → t¯. In general 4(0, t¯; U, u˜ n ) is too thin in C(0, t¯; E α ) to contribute much to the first coordinate of the set (5.24), thus M(t¯ ) has to be substantial. A closed set Z ⊆ Banach space X is T -full if for every sequence {xn } ⊆ Z such that xn → x¯ ∈ Z there exist ρ > 0 and a precompact sequence {Q n }, Q n ⊆ X , such that the intersection of all sets TZ (x n ) ∩ B(0, ρ) + Q n contains an interior point for n 0 large enough. Closed convex sets with nonempty interior are full; for other examples see [9], especially Lemma 5.2. Lemma 5.4. Assume that either (a) TM(t¯ ) ( y˜ n (·)) is T -full in C(0, t¯; E α ) and TY ( y˜ n ) is T -full in E α or (b) M(t¯ ) is T -full in C(0, t¯; E α ) and for every sequence {u˜ n (·)} ⊆ ¯ and every sequence { y˜ n } ⊆ Y with y˜ n → y(t¯, u) ¯ there Cad (0, t¯, U ) with u˜ n (·) → u(·) exists a precompact sequence {Q n }, Q n ⊆ E α , such that ∞ \

{R(0, t¯; U, u˜ n )(t¯ ) − TY ( y˜ n ) ∩ B(0, ρ) + Q n }

(5.25)

n=n 0

contains an interior point for n 0 large enough. Then the assumptions of Lemma 5.3 hold, so that any weak limit point (z 0 , µ, z) of the sequence (z 0n , µn , z n ) is nonzero. For the time optimal problem we replace R(0, t¯; U, u˜ n ) by R(0, t n ; U, u˜ n ) with tn < t¯, tn → t¯.

6.

The Navier–Stokes Equations

We apply the results to (1.1)–(1.4), or rather to the more general equation y0 (t) = A p y(t) + N (t, y(t), u(t)),

y(0) = ζ ,

(6.1)

with N (t, y, u) = Pp N (t, y, u), N (t, y, u)(x) = ϕ (t, x, y(x), ∇y(x), u(t)),

(6.2)

178


where ϕ = {ϕ1 , ϕ2 , . . . , ϕm } maps [0, T ] × Ä × Rm × Rm×m × U into Rm and ∇y = {(∂/∂ x k )yj } denotes the Jacobian matrix of y; below, ∇yϕ , ∇Yϕ also denote Jacobian matrices, the first with respect to y = {yj }, the second with respect to Y = {yjk }; ∇yϕ has m rows and m columns, whereas ∇Yϕ has m rows and m×m columns. The control set U is arbitrary. We assume the existence of an admissible control space Cad (0, t¯; U) such that for every u(·) ∈ Cad (0, t¯; U) the function ϕ (t, x, y, Y) = ϕ (t, x, y, Y, u(t)) is differentiable with respect to (y, Y) and that ∇(y,Y)ϕ (t, x, y, Y) = (∇yϕ (t, x, y, Y), ∇Yϕ (t, x, y, Y)) is continuous in y, Y for t, x fixed and measurable in t, x for y, Y fixed. Finally, we assume that for every c > 0 there exist K (·, c), L(·, c) ∈ L p (Ä) (independent of u(·)) such that |ϕ ϕ (t, x, y, Y)|Rm ≤ K (x, c), |∇(y,Y)ϕ (t, x, y, Y)|Rm ×Rm ×Rm×m ≤ L(x, c) m (6.3) (0 ≤ t ≤ T, x ∈ Ä, |y|R , |Y|Rm×m ≤ c). Theorem 6.1. Let p > m,

1 2

+ m/2 p < α < 1.

(6.4)

Then for every u(·) ∈ Cad (0, T ; U), N (t, y, u(t)) maps [0, T ]×(X p (Ä)m )α into X p (Ä)m and satisfies Hypothesis Dα,0 with constants K (c), L(c) independent of u(·); we have ∂y N (t, y, u(t)) = Pp ∂y N (t, y, u(t)), where ∂y N (t, y, u(t))h(x) = ∇yϕ (t, x, y(x), ∇y(x), u(t))h(x) + ∇Yϕ (t, x, y(x), ∇y(x), u(t))∇h(x)

(6.5)

(in the last term of (6.5) ∇h is ordered as a vector of length m × m). Proof. In view of (2.7) and of the first embedding (2.8), it is enough to consider the operator L(t, y, Y)(x) = ϕ (t, x, y(x), Y(x), u(t)) from C(Ä)m ×C(Ä)m×m into L p (Ä), where y and Y are uncoupled; for this operator the result is proved much in the same way as Lemma 7.1 in [14]. Theorem 6.1 guarantees that, if the admissible control set Cad (0, T ; U) is saturated and patch complete, all results in Section 5 for the model (2.22) can be applied to (6.1) after verification of closedness of the state constraint set M and the control set Y in E α . Regardless of the value of p and α, convergence in (X p (Ä)m )α = D((−A p )α ) implies L p convergence, thus pointwise convergence (of a subsequence) in Ä; hence, if inclusion is required a.e., the set M ⊆ (X p (Ä)m )α defined by the state constraint condition S y(t, x) ∈ M S in (1.15) is closed as required by Theorem 5.2 if M S is closed. Likewise, the target set Y ⊆ (X p (Ä)m )α defined by T y(t¯, x) ∈ MY is closed in in (X p (Ä)m )α if M S is closed. Remark 6.2. For a general control problem (resp. for the time optimal problem) Theorem 5.2 does not guarantee that the multiplier (z 0 , µ, z) is nonzero (resp. that the multiplier (µ, z) is nonzero). To achieve this we apply Lemma 5.4. We have D((−A p )α ) ⊂ → C (1) (Ä)m due to (2.7) and the embedding (2.8), since 2α > m/ p + 1 (condition (6.4)). This implies that if the coefficients of S and T are continuous in Ä and M S and MY


179

have nonempty interiors in their respective Euclidean spaces, then the sets M and Y have nonempty interiors in (X p (Ä)m )α . If M S and MY are in addition convex, M and Y will be convex as well, thus T -full (see Section 4). T -fullness survives cutting with a finite number of independent hyperplanes [13], thus we may supplement the target condition with a finite number of exact evaluations, for instance, E y(t¯, x¯ j ) = d j ∈ Rl , where E is a differential operator of the same form as T in (1.14). The same observation holds for the state constraint. Remark 6.3. There are constraints and target conditions important in practice that cannot be expressed by linear differential operators like (1.12) and (1.14). For instance, in constraints of helicity type, either integral or pointwise, |y(t, x) · (∇ · y(t, x))| ≤ C,

(6.6)

the operator is nonlinear. There is nothing in the treatment above that excludes nonlinear (1) m boundary conditions. The embedding D((−A p )α ) ⊂ → C (Ä) shows that the set M ⊆ p m α (X (Ä) ) defined by (6.6) is closed and has nonempty interior; however, it is no longer convex, thus application of the results would need a proof that the set M is T -full, something not attempted in this paper. For the Navier–Stokes nonlinearity (2.4) we have ∂y N = Pp ∂y N Pp with N (y) = −(y · ∇)y. The Fréchet derivative of N (y) is given by ∂y N (y)h = −(y · ∇)h − (h · ∇)y = −K(y)h − L(y)h. Coordinatewise, K(y)h = ((y · ∇)h)k = 6 j yj (∂/∂ x j )h k , so that (K(y)∗ z)k = −6 j (∂/∂ x j )(yj z k ) = −6 j yj (∂/∂ x j )z k = −((y · ∇)z)k , the second inequality stemming from the fact that y is divergence-free. On the other hand, (L(y)h)k = (6 j ((∂/∂ x j )yk )h j )k = ((∇y)h)k , hence L(y)∗ z = (∇y)T z. We finally obtain ∂y L(y)∗ z = (y · ∇)z − (∇y)T z.

(6.7)

Since ∂y N (y)∗ = Pq ∂y N (y)∗ Pq , the adjoint variational equation is the linear equation dz(s) = −Aq z(s) ds − Pq (y(s) · ∇)z(s) ds ¯ ds − µ(ds) + Pq (∇y(s))T z(s) ds − z 0 ∂y f 0 (s, y(s), u(s))

(6.8)

in the space X (Ä) , where µ ∈ 6(0, t¯; (X (Ä) )−α ) (note that ∂y f 0 (s, y(s), u(s)) ¯ ∈ X q (Ä)m ). For the time optimal problem, the minimum principle for (6.1) reads Z z¯ (s, x) · ϕ (s, x, y(s, x, u), ¯ ∇y(s, x, u), ¯ u(s)) ¯ dx Ä Z ¯ ∇y(s, x, u), ¯ v) d x (6.9) = min z¯ (s, x) · ϕ (s, x, y(s, x, u), q

v∈U

m

Ä

q

m

180


(note that (¯z, Pp ϕ) = (Pq z¯ , ϕ) = (¯z, ϕ). In particular, for the equation y0 (t) = A p y(t) + N (y(t)) + Pp u(t),

y(0) = ζ ,

(6.10)

with controls in the unit ball of L ∞ ((0, T ) × Ä)m , the minimum principle takes the form Z Z z¯ (s, x) · u(s, ¯ x) d x = min z¯ (s, x) · v(x) d x, (6.11) kvk L ∞ (Ä)m ≤1

Ä

Ä

which implies a “bang-bang” property for the components u¯ j (s, x) of the optimal control u(s, ¯ x) = (u¯ 1 (s, x), . . . , u¯ m (s, x)), u¯ j (s, x) = −sign z¯ j (s, x)

( j = 1, . . . , m),

(6.12)

wherever z¯ j (s, x) 6= 0. On the set where this happens see Remark 6.6. The result below, valid for (6.1), shows (as a very particular case) that the forcing measure µ(ds) in the adjoint variational equation is zero in case there are no state constraints, that is, when M = (X p (Ä)m )−α . Lemma 6.4. Let µ ∈ 6(0, t¯; (X q (Ä)m )−α ). Then if µ satisfies the second condition (5.7), ∇ × µ = 0 in the set ¯ ∈ Int(M S )}. e0 = {(t, x) ∈ [0, t¯ ] × Ä; Sy(t, x, u)

(6.13)

Since µ itself is not a measure, the statement must be clarified. It means Z A pϕ (t, x) · ν (dt d x) = 0, [0,t¯ ]×Ä

for any divergence-free test vector ϕ with support contained in an arbitrary open set B ⊆ R × Rm with B ∩ ([0, t¯ ] × Ä) ⊆ e0 , where ν = ((−A p )∗ )−1µ = (−Aq )−1µ = (−Aq )α−1 (−Aq )−αµ. 1 m Note that (−Aq )−αµ ∈ 6(0, t¯; (X q (Ä)m )) ⊂ → 6(0, t¯; L (Ä) ) ⊂ → 6(0, t; 6(Ä)) is a bona fide measure, that is, an element of the space 6([0, t¯ ] × Ä) = 6(0, t¯; 6(Ä)) of all regular, bounded Borel measures in the cylinder [0, t¯ ] × Ä; a fortiori, so is ν . The proof is similar to that of Lemma 3.3 in [14] and we omit it (the fact that the operator A p is not local plays no role). In case there are no state constraints, e0 = [0, t¯ ] × Ä and µ = 0. There is an obvious analogue of Lemma 6.4 for the target condition (1.13).

Remark 6.5. If the differential operators S and T in the state constraints involve only m y and not its derivatives, we only need the embedding D((−A p )α ) ⊂ → C(Ä) . This only requires α > m/2 p with p > m/2 so that we can use Theorem 2.1 instead of Theorem 6.1; for m = 2, 3 we may base the treatment on the space X 2 (Ä)m with α > 12 (m = 2) or α > 34 (m = 3). For α = 12 we cannot ensure nontriviality of the multipliers for pointwise constraints, but we can still consider state constraints of integral form (0 ≤ t ≤ t¯ ),

S y(t, ·) ∈ M S

(6.14)

where M ⊆ L (Ä) is convex with interior points, and target conditions of the same type. 2

k


181

Remark 6.6. Results such as the “bang-bang theorem” (6.12) lead to the following problem. Let µ ∈ 6(0, t¯; X q (Ä)−α ), and let z(t, x) = (z 1 (s, x), . . . , z m (s, x)) be a nontrivial solution of the adjoint variational equation (6.8) (“nontrivial” means “not identically zero”). What can be said about the sets e j (t¯, z, µ) = {(t, x) ∈ [0, t¯ ] × Ä; z j (t, x) = 0}?

(6.15)

In the minimum principle the final condition z¯ (t¯, ·) belongs to X q (Ä)−α ; note, however, that z¯ (t, ·) is smoothed into X q (Ä)m for t < t¯, thus we may assume that z¯ (t¯, ·) belongs to X q (Ä)m (or even to smaller spaces). We do not know of any results on this problem, even in the case where µ = 0.

7.

Point Targets, I

The model here is (2.12), with bounded control set U ⊆ X ∗ and Cad (0, T ; U ) defined as in Section 2, and the application in mind is y0 (t) = A p y(t) + N (y(t)) + Pp u(t),

y(0) = ζ ,

(7.1)

with N (y) = Pp L(y),

L(y) = ϕ (x, y(t, x), ∇y(t, x)),

(7.2)

where ϕ = {ϕ1 , . . . , ϕm } maps Ä × Rm × Rm×m into Rm (a t-dependent ϕ can be handled but we omit this case for simplicity). We assume that ϕ (x, y, Y) is continuously differentiable with respect to x ∈ Ä, y ∈ Rm , Y ∈ Rm×m , the derivatives and ϕ locally Lipschitz continuous in Ä × Rm × Rm×m . All of these conditions are satisfied by the Navier−Stokes nonlinearity (2.4). Lemma 7.1. Let α − 1/2 p −

1 2

1 2

< α < 1, p > m so large that

> m/ p.

(7.3)

Then there exists δ > 0 such that the operator N defined in (7.2) maps (X p (Ä)m )α into (X p (Ä)m )δ and it has a Fréchet derivative ∂y N (y) ∈ ((X p (Ä)m )α , (X p (Ä)m )δ ). N (y) (resp. the Fréchet derivative ∂y N (y)) is continuous and locally bounded from (X p (Ä)m )α into (X p (Ä)m )δ (resp. from (X p (Ä)m )α into ((X p (Ä)m )α , (X p (Ä)m )δ )). Proof.

The left side of (7.3) is positive, so that α −

0 < δ < 1/2 p,

ε = (α − δ)/2 −

1 4

1 2

> 1/2 p. Select δ such that

> m/2 p.

We have 2α − 2ε − 1 = 2δ + 2ε = s > 2ε > m/ p. It follows from the first embedding (2.11) and the first line of (2.7) that D((−A p )α ) ≈ X p (Ä)m ∩ H0

2α, p

2α−2ε, p

p m (Ä)m ⊂ → X (Ä) ∩ W0

(Ä)m .

(7.4)

182


On the other hand we have 2δ < 1/ p, thus we use the second embedding (2.11) and the second line of (2.7): p m 2δ, p (Ä)m = D((−A p )δ ). X p (Ä)m ∩ W 2δ+2ε, p (Ä)m ⊂ → X (Ä) ∩ H

(7.5)

It then follows that to prove Lemma 7.1 it is enough to prove the four lemmas below. In the first, the scalar function ψ(x, y1 , . . . , yn ) is continuously differentiable with respect to x ∈ Ä, yj ∈ R, the derivatives and ψ locally Lipschitz continuous in Ä × Rn . Lemma 7.2.

Let 0 < s < 1, s > m/ p. The operator

9(y1 (·), . . . , yn (·))(x) = ψ(x, y1 (x), . . . , yn (x))

(7.6)

maps W s, p (Ä) × · · · × W s, p (Ä) into W s, p (Ä) and is continuous and locally bounded, with a continuous locally bounded Fréchet derivative with respect to y given by ∂9(y1 , . . . , yn )(h 1 , . . . , h m ) =

n X ∂ψ (x, y1 (x), . . . , yn (x))h j (x). ∂ xj j=1

(7.7)

Lemma 7.3. Let 0 < s < 1 and s > m/ p. Then kyzkW s, p (Ä) ≤ CkykW s, p (Ä) kzkW s, p (Ä) Lemma 7.4. let

(y, z ∈ W s, p (Ä)).

(7.8)

Let ψ(x, y1 , . . . , y n ) be locally Lipschitz continuous in all variables, and

9(y1 , . . . , yn )(x) = ψ(x, y1 (x), . . . , yn (x)).

(7.9)

Then 9 is a locally bounded, continuous operator from W s, p (Ä) × · · · × W s, p (Ä) into W s, p (Ä). Lemma 7.5. Let 0 < s < 1. Then Pp is a bounded operator from W s, p (Ä) into itself. Lemmas 7.2 and 7.3 are classical (for a proof see Lemmas 8.2 and 8.3 of [14]). For Lemma 7.5 see (0.2.21) on p. XXIII of [47]. We consider the abstract model (2.12), y 0 (t) = Ay(t) + f (t, y(t)) + Bu(t),

y(0) = ζ,

(7.10)

under the requirements in Section 2, except that we assume that f (t, y) ∈ D((−A)δ ) and that (−A)δ f (t, y) satisfies Hypothesis Dα,0 Lemma 7.6. Let u¯ ∈ Cad (0, t¯; U ) be such that y(t, u) ¯ exists in 0 ≤ t ≤ t¯ and y(t¯, u) ¯ ∈ ¯ p, v) ∈ E 1 , the sets {e j (h)} in Theorem 4.4 E 1 = D(A). Then, if v is such that ξ(t, u, can be chosen in such a way that the conclusions there are true, and, in addition, y(t¯, u¯ e(h),v ) ∈ E 1 and ξ(t, u, ¯ p, v) = lim

h→0+

in the norm of E 1 .

1 (y(t, u¯ e(h),v ) − y(t, u)) ¯ h

(7.11)


183

The proof is essentially the same as that of Lemma 8.5 of [14]. It is also shown there that the condition that y(t¯, u) ∈ D(A) is equivalent to u(·) ∈ D(A, t¯ ), the subspace of ∗ L∞ w (0, T ; X ) defined by Z

t¯

S(t¯ − τ )Bu(τ ) dτ ∈ D(A).

(7.12)

0

Accordingly, if y(t¯, u) ¯ ∈ D(A), then ξ(t¯, u, ¯ p, v) ∈ D(A) if and only if D(A, t¯ ).

8.

P

p j v j (·) ∈

Point Targets, II

The proof of the minimum principle is very similar to that in [14] and we limit ourselves to sketching it. We apply again the nonlinear programming theory in Section 4 to the ¯ δ) ⊆ B(u, ¯ δ) of the ball in functions (4.19). The metric space V is the subspace B1 (u, Lemma 2.2 consisting of all u(·) ∈ B(u, ¯ δ) with y(t¯, u) ¯ ∈ E 1 = D(A), equipped with the distance ρ(u(·), v(·)) = d(u(·), v(·)) + ky(t¯, u) − y(t¯, v)k E1 ,

(8.1)

where d is the distance (2.22); B1 (u, ¯ δ) is complete under this norm and f, as a func¯ δ); likewise, tion with values in C(0, t¯; E α ) × E 1 , is defined and continuous in B1 (u, ¯ δ). Since d ≤ ρ, a directhe R-valued function f0 is defined and continuous in B1 (u, ¯ δ), ρ) is also a directional derivative of f in tional derivative (ξ(·), ξ(t¯ )) of f in (B1 (u, ¯ δ), d); in the opposite direction, we check that if (ξ(·), ξ(t¯ )) ∈ C(0, t¯; E α )×E 1 is (B1 (u, ¯ δ), d) and c > kξ(t¯ )k E1 +1, then c−1 (ξ(·), ξ(t¯ )) a directional derivative of f at u˜ in (B1 (u, is a directional derivative of f at u˜ in (B1 (u, ¯ δ), ρ). In the minimum principle below, the cost functional is (5.2); f 0 (t, y, u(t)) satisfies Hypothesis Dα0 in Section 5 for all u(·) ∈ Cad (0, T ; U ) with K 0 (c), L 0 (c) independent of u(·). The state constraint and target conditions are (5.1) with M closed in E α and Y closed in E 1 . The control space Cad (0, T ; U ) is saturated and patch complete; we also assume that zero is an interior point of U in X ∗ . The adjoint semigroup S(t)∗ is extended to the spaces (E ∗ )−α by S(t)∗ = ((−A)α )∗ S(t)∗ ((−A)−α )∗

(t > 0).

(8.2)

We denote by (E ∗ )1−1 (B ∗ ) the subspace of all z ∈ (E ∗ )−1 with Z

1

kB ∗ S(t)∗ zk X dt =

Z

0

1

kB ∗ (−A∗ )S(t)∗ (−A∗ )−1 zk X dt < ∞.

(8.3)

0

Theorem 8.1. Let u(·) ¯ ∈ Cad (0, t¯; U ) be an optimal control such that y(t¯, u) ¯ ∈ E1 = D(A). Then there exists (z 0 , µ, z) ∈ R × 6(0, t¯; (E∗)−α ) × (E ∗ )1−1 (B ∗ ) with z 0 ≥ 0,

³ ´− µ ∈ lim inf TM(t¯ ) ( y˜ n (·)) , n→∞

³ ´− z ∈ lim inf TY ( y˜ n ) , n→∞

(8.4)

184


where { y˜ n (·)} (resp. y˜ n ) is a sequence in M(t¯ ) (resp. Y ) such that y˜ n (·) → y(·, u) ¯ in ¯ in E 1 ) and such that if z¯ (s) is the solution of C(0, t¯; E α ) (resp. y˜ n → y(t¯, u) ¯ ∗ }¯z (s) ds d z¯ (s) = −{A∗ + ∂ y f (s, y¯ (s, u)) ¯ u(s)) ¯ ds − µ(ds), − z 0 ∂ y f 0 (s, y(s, u),

z¯ (t¯ ) = z,

(8.5)

then Z

t¯

z0

{ f 0 (s, y(s, u), ¯ v(s)) − f 0 (s, y(s, u), ¯ u(s))} ¯ ds

0

Z

t¯

+

hB ∗ z¯ (s), v(s) − u(s)i ¯ ds ≥ 0

(8.6)

0

for all v(·) ∈ Cad (0, t¯; U ). Proof. We apply Theorem 4.1 in the space B1 (u, δ) with target set Y = M(t¯ ) × Y ⊆ C(0, t¯; E α ) × E 1 , obtaining the sequences { y˜ n (·), y˜ n } ⊆ M(t¯ ) × Y in the statement and a sequence u˜ n (·) ⊆ B1 (u, ¯ δ) with ρ(u˜ n , u) ¯ = d(u˜ n , u) ¯ + ky(t¯, u˜ n ) − y(t¯, u)k ¯ E1 → 0. n We use the sequence Dn , Dn ⊆ Der(f0 , f)(u˜ ), of convex sets consisting of all triples (t¯−1 ξ0 (t¯, u˜ n , p, v), t¯−1 ξ(·, u˜ n , p, v), t¯−1 ξ(t¯, u˜ n , p, v))

(8.7)

with 6p j v j (·) ∈ D(A, t¯ ), so that ξ(t¯, u˜ n , p, v) ∈ D(A) (since y(t¯, u˜ n ) ∈ D(A), u˜ n (·) ∈ ¯ → 0 and the integral equations satisfied by D(A, t¯ )). Using the fact that ρ(u˜ n , u) y(t, u), ¯ y(t, u˜ n ) we obtain Z

t¯

(−A)

S(t¯ − τ )B(u˜ n (τ ) − u(τ ¯ )) dτ → 0

as n → ∞,

(8.8)

0

and then use the following result. Lemma 8.2.

Let 6p j v j (·) ∈ D(A, t¯ ). Then ξ(t¯, u˜ n , p, v) → ξ(t¯, u, ¯ p, v) in E 1 .

The proof follows from the convergence relation (8.8) and the integral equations satisfied by ξ(t¯, u˜ n , p, v) and ξ(t¯, u, ¯ p, v). End of Proof of Theorem 8.1. Theorem 4.1 gives sequences {z 0n } ⊆ R, µn ⊆ 6(0, t¯; (E ∗ )−α ), and {z n } ⊆ (E ∗ )−1 such that 2 z 0n + kµn k26(0,T ;(E ∗ )−α ) + kz n k2(E ∗ )−1 = 1, z 0n ≥ 0, n z 0n ξ0 (t¯, u˜ , p, v) + hµn , ξ(·, u˜ n , p, v)ic,α + hz n , ξ(t¯, u˜ n , p, v)i1 ≥ −δn ,

(8.9)

δn → 0, where the second angled bracket indicates the duality of the spaces E 1 and (E ∗ )−1 . If (z 0 , µ, z) is a (R × C(0, t¯; E α ) × E 1 )∗ -weak limit point of (z 0n , µn , z n ), then we have ¯ p, v) + hµ, ξ(·, u, ¯ p, v)ic,α + hz, ξ(t¯, u, ¯ p, v)i1 ≥ 0. z 0 ξ0 (t¯, u,

(8.10)


185

For a single patch with v(·) ∈ D(A, t¯ ), ¯ v) + hµ, ξ(·, u, ¯ v)ic,α + hz, ξ(t¯, u, ¯ v)i1 ≥ 0, z 0 ξ0 (t¯, u,

(8.11)

or Z

t¯

z0


0

+ z0

Z t¯ ¿

¯ u(τ ¯ )), ((−A)−α )∗ ∂ y f 0 (τ, y(τ, u), À Z τ Rα,0 (τ, σ, u){ ¯ f (σ, y(σ, u), ¯ v(σ )) − f (σ, y(σ, u), ¯ u(σ ¯ ))} dσ dτ

0

+

0

Z t¯ ¿

((−A)−α )∗ µ(dt), À Z t Rα,0 (t, σ ; u){ ¯ f (σ, y(σ, u), ¯ v(σ )) − f (σ, y(σ, u), ¯ u(σ ¯ ))} dσ

0

0

¿

+ ((−A)−1 )∗ z, Z (−A)

1−α

Z = z0 0

+ z0

À Rα,0 (t¯, σ ; u){ ¯ f (σ, y(σ, u), ¯ v(σ )) − f (σ, y(σ, u), ¯ u(σ ¯ ))} dσ

0 t¯


Z t¯ ¿ 0

+

t¯

¯ u(τ ¯ )), ((−A)−α )∗ ∂ y f 0 (τ, y(τ, u), À Z τ Rα,0 (τ, σ, u)B(v(σ ¯ ) − u(σ ¯ )) dσ dτ 0

Z t¯ ¿

−α ∗

((−A)

Z

t

) µ(dt),

À Rα,0 (t, σ ; u)B(v(σ ¯ ) − u(σ ¯ )) dσ

0

0

¿ Z + ((−A)−1 )∗ z, (−A)1−α

t¯

À Rα,0 (t¯, σ ; u)B(v(σ ¯ ) − u(σ ¯ )) dσ ≥ 0. (8.12)

0

To obtain the minimum principle we need to pull operators to the left side of all duality products, which we do as in Theorem 11.2.2 in all terms except the last, Z

t¯

(−A)1−α

Rα,0 (t¯, σ ; u)B(v(σ ¯ ) − u(σ ¯ )) dσ.

(8.13)

0

¯ (see the details in [14]) we obtain Using (3.7) for Rα,0 (t, s; u) (−A)1−α Rα,0 (t, s; u) ¯ = (−A)S(t − s) + N (t, s; u), ¯

(8.14)

with N (t, s; u) ¯ continuous in the norm of (E, E) for s < t and kN (t, s; u)k ¯ (E,E) ≤ C(t − s)κ (t > s) with κ < 1. It then results that, since v(·) ∈ D(A, t¯ ) the operators

186


may be pulled to the other side of the duality product. All the other terms in (8.12) remain bounded for arbitrary v(·) ∈ D(A, t¯ ), hence Z t¯ h(−A∗ )S(t¯ − σ )∗ (−A∗ )−1 z, B(v(σ ) − u(σ ¯ ))i dσ 0

¿

∗ −1

Z

= (−A ) z,

t¯

À (−A)S(t¯ − σ )B(v(σ ) − u(σ ¯ )) dσ

(8.15)

0

is bounded below for v(·) ∈ D(A, t¯ ). This and the fact that U contains a ball around the origin in H implies that (8.3) holds, so that z ∈ (E ∗ )1−1 (B ∗ ) as claimed, and we may then use an arbitrary control v(·) ∈ Cad (0, t¯; U ) whether or not it belongs to D(A, t¯ ). This ends the proof of Theorem 8.1. The time optimal problem needs separate treatment. The functions {fn } in the abstract time optimal problem in Section 4 are fn (u) = (y(·, u), y(tn , u)) ∈ C(0, t¯; E α )× E 1 , {tn } ¯ δ) consisting a sequence in [0, t¯ ) (t¯ = optimal time), each fn defined in the space Bn (u, ¯ δ) is of all u ∈ B(u, ¯ δ) such that y(tn , u) ∈ D(A) = E 1 ; the distance dn in Bn (u, (8.1) with tn instead of t¯. The target set is the same, and it is obvious that, due to time ¯ δ)) ∩ Y = ∅. We extend u(·) ¯ by u(t) ¯ = u( ¯ t¯ ) for t ≥ t¯ and define a optimality, fn (Bn (u, sequence {u¯ n (·)} ⊆ Bn (u, ¯ δ) by u¯ n (t) = u(t + (t¯ − tn )) (0 ≤ t ≤ t¯ ). Lemma 8.3. ¯ in C(0, t¯; E α ). (a) y(·, u¯ n ) → y(·, u) n ¯ in E 1 . (b) y(tn , u¯ ) ∈ E 1 and y(tn , u¯ n ) → y(t¯, u) The proof is the same as that of Lemma 8.1 in [14]. Lemma 8.3 certifies in particular that dist(f(u¯ n ), Y) → 0 as n → ∞ , so that {u¯ n (·)} is an optimal sequence. Using Theorem 4.2 we obtain the “abstract time optimal” analog of (8.9), kµn k26(0,T ;(E ∗ )−α ) + kz n k2(E ∗ )−1 = 1, hµn , ξ(·, u˜ n , p, v)ic,α + hz n , ξ(t¯, u˜ n , p, v)i1 ≥ −δn ,

(8.16)

with δn → 0. To pass to ¯ p, v)ic,α + hz n , ξ(t¯, u, ¯ p, v)i1 ≥ 0 hµn , ξ(·, u,

(8.17)

we must show that ¯ p, v), ξ(t¯, u, ¯ p, v)) (ξ(·, u˜ n , p, v), ξ(tn , u˜ n , p, v)) → (ξ(·, u,

(8.18)

in C(0, t¯; E α ) × E 1 . This does not follow directly from Lemma 8.3; however, since ρn (u˜ n , u¯ n ) → 0 it is enough to show (8.18) with u¯ n instead of u˜ n . Using the integral equations for (−A)α ξ(t, u¯ n , p, v) and (−A)α ξ(t, u¯ n , p, v), convergence of the first co¯ in L r (0, t¯; E). Convergence of ordinate of (8.18) reduces to the fact that u¯ n (·) → u(·) the second coordinate in (8.18) is handled as in Lemma 8.3. In the result below nothing is asked from TY ( y˜ n ) so that the point target condition y(t¯, u) = y¯ ∈ D(A) is included.


187

Lemma 8.4. Assume that X ∗ = E, B = I. Then the multiplier (z 0 , µ, z) is nontrivial; for the time optimal principle, the multiplier (µ, z) is nontrivial. Proof. We apply Lemma 5.4. In the time optimal case all we have to show is that the sets R(0, tn ; U, u˜ n ) contain a common ball in D(A) for n ≥ n 0 ; if this is achieved, we do not need any help from TY ( y˜ n ) in (5.23)–(5.24) or from the precompact sequence {Qn } (we may take Qn = {0} for all n). Since R(0, tn ; U, u˜ n ) = {ξ(tn ; u˜ n , v); v(·) ∈ Cad (0, t n ; U )}

(8.19)

¯ v), it is enough to prove that and, for fixed v, we have ξ(tn ; u˜ n , v) → ξ(t¯; u, {ξ(tn ; 0, v); v(·) ∈ Cad (0, tn ; U )}

(8.20)

contains an interior point in D(A) for n large enough. To see this note that if y ∈ D(A) and y(t) = t y, then y(t) satisfies y 0 (t) = y = {A + ∂ y f (t, y(t, u˜ n (t)))}y(t) + {y − t Ay − t∂ y f (t, y(t, u˜ n ))y} y(0) = 0, (8.21) = {A + ∂ y f (t, y(t, u˜ n (t)))}y(t) + v(t), and we obviously have kv(t)k E ≤ Ckyk D(A) (0 ≤ t ≤ t¯ ). This ends the proof. Remark 8.5. The results are applicable to (7.1); we take X = L q (Ä)m , X ∗ = L p (Ä)m , so that Cad (0, T ; U ) is a subspace of L ∞ (0, T ; L p (Ä)m ). The conditions p > 2m, α > 1 + m/ p imply the corresponding conditions on p, m, α in Lemma 7.1, thus all the 2 comments in Remark 6.2 on the state constraints apply. However, we do not need any conditions on the coefficients of T , and it is enough to require MY to be closed.

9.

Combustion Processes

The combustion model (1.18)–(1.21) can be handled the same as the Navier−Stokes equations. The space is E = X p (Ä)m × L p (Ä)n endowed with its p-product norm and the equations become the system y0 (t) = A p y(t) − Pp (y · ∇)y + Pp u(t),

(9.1)

ψ 0 (t) = 1 pψ (t) − (y · ∇)ψ ψ + f(ψ ψ ) + v(t),

(9.2)

with y(t)(x) = y(t, x), ψ (t)(x) = ψ (t, x), u(t)(x) = u(t, x), v(t)(x) = v(t, x), y(t, x) = (y1 (t, x), . . . , ym (t, x)),

ψ (t, x) = (ψ1 (t, x), . . . , ψn (t, x)),

u(t, x) = (u 1 (t, x), . . . , u m (t, x)),

v(t, x) = (v1 (t, x), . . . , vn (t, x)),

and f is the nonlinear vector function f(ψ ψ ) = f(ψ1 , . . . , ψn ) = ( f 1 (ψ1 , . . . , ψn ), . . . , f n (ψ1 , . . . , ψn )),

188


A p is the Stokes operator in X p (Ä)m , and 1 pψ = (ν1 1ψ1 , . . . , νn 1ψn ). Since A p (resp. 1 p ) generates a bounded analytic semigroup in X p (Ä)m (resp. L p (Ä)n ), the operator · ¸ Ap 0 (9.3) Ap = 0 1p 2, p

2, p

with domain D(A p ) = (X p (Ä)m ∩ W0 (Ä)m ) × W0 (Ä)n generates a bounded analytic semigroup in X p (Ä)m × L p (Ä)n . Combining (2.6) with (2.7), the domains of the fractional powers are given by ( 2α, p 2α, p (X p (Ä)m ∩ H0 (Ä)m ) × H0 (Ä)n (1/2 p < α < 1), α D((−A p ) ) ≈ (9.4) (X p (Ä)m ∩ H 2α, p (Ä)m ) × H 2, p (Ä)n (0 < α < 1/2 p). The nonlinearity in system (9.1) is ¸ · ¸ · y −Pp (y · ∇)y . N = −(y · ∇)ψ ψ + f(ψ ψ) ψ

(9.5)

Using the same techniques as in Theorem 6.1 we show that if the f j are continuously differentiable, then N satisfies Hypothesis Dα,0 when p, m, α verify (6.4). Further, if the derivatives are locally Lipschitz continuous, then the argument in Lemma 7.3 can be used to show that N : D((−A p )α ) → D((−A p )δ ) for p, m, α, δ as in (7.3), and that N with respect to y, ψ is given (−A p )−δN satisfies Hypothesis Dα,0 . The differential ∂N by · ¸· ¸ · ¸ y h −Pp (y · ∇)h − Pp (h · ∇)y ∂N N = ψ υ −(h · ∇)ψ ψ − (y · ∇)υ υ + (∇f(ψ ψ ))υ υ ¸· ¸ · h 0 −Pp K(y) − Pp L(y) . (9.6) = −L(ψ ψ) −K(y) + ∇f(ψ ψ) υ State constraints and target conditions for the system (9.1)–(9.2) are formulated on the basis of differential operators like (1.11) and (1.13) but involving also the ψ j . For the formulation of the minimum principle we need the adjoint of ∂N N , given by the formula ¸· ¸ · ¸∗ · ¸ · z z y −(∇ψ ψ )T Pp (y · ∇) − Pp (∇y)T = . (9.7) ∂N N T 0 (y · ∇) + (∇f(ψ ψ )) υ υ ψ Hence, taking into account that the operator A∗p : X q (Ä)m × L q (Ä)n → X q (Ä)m × L q (Ä)n is given by · ∗ Ap ∗ Ap = 0

¸ · 0 Aq = 1∗p 0

¸ 0 , 1q

(9.8)

the adjoint variational equation is the system dz(s) = −Aq z(s) ds − Pp (y(s) · ∇)z(s) ds ψ )T υ (s) − µ(ds), + Pp (∇y(s))z(s) ds + (∇ψ υ (s) ds − (∇f(ψ ψ (s))T υ (s) ds − ν (ds), dυ υ (s) = 1q υ (s) ds − (y(s) · ∇)υ

(9.9) (9.10)


189

where µ ∈ 6(0, t¯; (X q (Ä)m )−α ) and ν ∈ 6(0, t¯; (L q (Ä)n )−α ). Taking as control space the unit ball of L ∞ (Ä)m × L ∞ (Ä)n the minimum principle for a time optimal control (u(·), ¯ v(·)) ¯ reads Z {¯z(s, x) · u(s, ¯ x) + υ¯ (s, x) · v¯ (s, x)} d x Ä Z {¯z(s, x) · u(x) + υ¯ (s, x) · v(x)} d x. (9.11) = min kuk L ∞ (Ä)m ≤1, kvk L ∞ (Ä)n ≤1 Ä

All that was said in Section 6 about the Navier–Stokes equations applies here, in particular Remark 6.2 on nontriviality of the multipliers and Lemma 6.4 on saturation of the state constraints. The material on point targets in Sections 7 and 8 also applies, with X ∗ = L p (Ä)m × L p (Ä)n ; see in particular Remark 8.5.

10.

Existence of Optimal Controls

We examine the question for the model (2.12). All assumptions in Section 2 are in force; in addition we require the semigroup S(t) to be compact. Existence results are based on ∗ Theorem 10.1. Let 0 < κ < 1. Then the operator 3κ : L ∞ w (0, T ; X ) → C(0, T ; E) defined by Z t (−A)κ S(t − τ )Bu(τ ) dτ (0 ≤ t ≤ t¯ ) (10.1) 3κ u(·) = 0

is compact. The proof corresponding to κ = 0 is in [10, Theorem 3.1]; the proof for κ > 0 is similar. We consider a general cost functional y0 (t, u) for (2.12) (not necessarily of the form (5.2)). We say that y0 (t, u) is weakly lower semicontinuous if y(t¯, u) ¯ ≤ lim sup y0 (t¯, u n )

(10.2)

n→∞

¯ ∗ for every sequence u n (·) ⊆ L ∞ ¯ L 1 (0, t¯; U )-weakly in w (0, t ; X ) with u n (·) → u(·) ∞ ∗ n ¯ ¯ L w (0, t ; X ) and such that y(t, u), ¯ y(t, u ) exist in 0 ≤ t ≤ t and y(t, u n ) → y(t, u) ¯ ¯ uniformly in 0 ≤ t ≤ t . The cost functional is equicontinuous if y0 (tn , u n )− y0 (t¯, u n ) → ∗ ¯ ¯ L 1 (0, t˜; U )0 for every sequence u n (·) ⊆ L ∞ w (0, tn ; X ) such that tn → t , u n (·) → u(·) ∞ ∗ ˜ weakly in L w (0, t ; X ), y(t, u n ) exists in 0 ≤ t ≤ tn and is uniformly bounded there (t˜ > tn for all n and u n (·) is extended by u n (t) = u fixed in tn ≤ t ≤ t˜). Let m be the minimum of the functional y0 (t¯, u) for all u(·) ∈ Cad (0, t¯; U ) such that y(t, u) exists in 0 ≤ t ≤ t¯ and satisfies the state constraints and the target condition; we assume that −∞ < m < ∞, and the terminal time t¯ may be fixed or free. A sequence {u n (·)}, u n ∈ Cad (0, tn ; U ), is a minimizing sequence if y(t, u n ) exists in 0 ≤ t ≤ tn and lim dist(y(t, u n ), M) = 0

n→∞

lim dist(y(tn , u n ), Y ) = 0,

n→∞

(0 ≤ t ≤ lim inf tn ), (10.3)

190


lim inf y0 (tn , u n ) ≤ m, n→∞

(10.4)

the distance in (10.3) that of E α . We do not require the sequence {tn } to be bounded, although this assumption is added in Theorem 10.2. In the result below, we assume that the state constraint set M and the target set Y in (5.1) are closed in E α , and that for each t¯ the control space Cad (0, t¯; U ) is L 1 (0, t¯; X )∗ weakly compact in L ∞ w (0, T ; X ) (by Alaoglu’s theorem, this holds if U is the unit ball of X ∗ ). Theorem 10.2. Assume there exists a minimizing sequence u n (·) with tn bounded and y(t, u n ) uniformly bounded. Then there exists an optimal control. The proof follows a familiar pattern (see [10]) thus we only sketch it. We begin by taking t˜ > tn for all n and extending u n (·) and y(·, u n ) to [0, t˜ ] setting u n (t) = u = fixed element of U in tn ≤ t ≤ t˜, and y(t, u n ) = y(tn , u n ) in tn ≤ t ≤ t˜. Define y˜ (t, u n ) in 0 ≤ t ≤ t˜ as follows: y˜ (t, u n ) = y(t, u n ) for 0 ≤ t ≤ tn . For t > tn , y˜ (t, u n ) is defined by (2.18) with the previously extended y(t, u n ) on the right side. Next, we take a subsequence ˜ ∗ ¯ ∈ Cad (0, t¯; U ) ⊆ L ∞ of u n (·) such that tn → t¯ and that u n (·) → u(·) w (0, t ; X ) α L 1 (0, t˜; X )-weakly. Finally, we write the integral equation (2.18) for (−A) y(t, u n ) and use Theorem 10.1 to show that, if need be taking a subsequence, (−A)α y(t, u n ) is uniformly convergent in 0 ≤ t ≤ t¯; taking limits in the integral equation we see ¯ It follows from weak lower semicontinuity and that y(t) = limn→∞ y(t, u n ) = y(t, u). ¯ ≤ m. In view of the first condition equicontinuity of y0 (t, u) and from (10.2) that y(t¯, u) (10.3) we have y(t, u) ∈ M (0 ≤ t ≤ t¯ ). The fact that the target condition holds follows from the facts that y(t¯, u n ) = y(tn , u n ) + (y(t¯, u) − y(tn , u n )) and from Theorem 10.1 noting that uniform convergence implies equicontinuity. Theorem 10.2 can be applied straight to the Navier–Stokes equations (2.3) with control space L r (Ä)m , p ≤ r ≤ ∞; X = L s (Ä) (1/r + 1/s = 1). The same applies to the combustion model (9.1)–(9.2). Existence theory for the model (2.22) is far more delicate, since control may appear nonlinearly, and use of relaxed controls may be necessary; see [10] for results in that direction.

References 1.

Abergel E, Temam R (1990) On some control problems in fluid mechanics. Theoret Comput Fluid Mech 1:303–325 2. Adams RA (1975) Sobolev Spaces. Academic Press, New York 3. Amann H, Hieber M, Simonett G (1994) Bounded H ∞ -calculus for elliptic operators. Differential Integral Equations 7:1–64 4. Buckmaster J, Ludford GSS (1982) Theory of Laminar Flames, Cambridge University Press, New York 5. Casas E. (1992) Optimal control in coefficients of elliptic equations with state constraints. Appl Math Optim 26:21–37 6. Casas E (1994) The Navier–Stokes equations coupled with the heat equation: analysis and control. Control and Cybernet 23:605–620 7. Casas E, Fernández LA (1993) Optimal control in coefficients of semilinear elliptic equations with pointwise state constraints on the gradient of the state. Appl Math Optim 27:35–36

Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

19. 20. 21. 22.

23. 24. 25. 26. 27.

28. 29. 30. 31. 32. 33. 34. 35.

191

Diestel JJ, Uhl JJ (1977) Vector Measures. American Mathematical Society, Providence, RI Fattorini HO (1993) Optimal control problems in Banach spaces. Appl Math Optim 28:225–257. Fattorini HO (1993) Relaxation in infinite dimensional control systems. L. Markus Festschrift Volume, pp 505–522 (KD Elworthy, WN Everitt, EB Lee, eds). Marcel Dekker, New York Fattorini HO (1994) Existence theory and the maximum principle for relaxed infinite dimensional optimal control problems. SIAM J Control Optim 32:311–331 Fattorini HO (1995) The maximum principle for linear infinite dimensional control systems with state constraints. Discrete Cont Dynam Systems 1:77–101 Fattorini HO (1996) Optimal control problems with state constraints for semilinear distributed parameter systems. J Optim Theory Appl 88:25–59 Fattorini HO (to appear) Optimal control problems with state constraints for distributed parameter systems; the parabolic case. Fattorini HO, Frankowska H (1991) Infinite dimensional control problems with state constraints. Lecture Notes in Control and Information Sciences, vol 154, pp 52–62. Springer-Verlag, Berlin Fattorini HO, Sritharan SS (1992) Existence of optimal controls for viscous flow problems. Proc Roy Soc London Ser A 439:81–102 Fattorini HO, Sritharan SS (1994) Necessary and sufficient conditions for optimal controls in viscous flow problems. Proc Roy Soc Edinburgh Sect A 124:211–251 Fattorini HO, Sritharan SS (1994) Relaxation in semilinear infinite dimensional systems modelling fluid flow control problems. Control and Optimal Design of Distributed Parameter Systems, pp 93–111 (DL Russell, J Magnese, L White, eds). Springer-Verlag, New York Fattorini HO, Sritharan SS (1995) Optimal chattering controls for viscous flow. Nonlinear Anal TMA 25:763–797 Frankowska H (1990) Some inverse mapping theorems. Ann Inst H Poincaré 7:183–234 Fujiwara D, Morimoto H (1977) An L r -theorem of the Helmholtz decomposition of vector fields. J Fac Sci Univ Tokyo I:685–700 Fursikov AV (1981) On some control problems and results related to the unique solution of mixed problems for the three-dimensional Navier–Stokes equations and Euler equations. Dokl Akad Nauk SSSR 89:1066–1070 Fursikov AV (1981) Control problems and results on the unique solution of mixed problems for the three-dimensional Navier–Stokes and Euler equations. Mat Sb 115:281–306 Giga Y (1981) Analyticity of the semigroup generated by the Stokes operator in L r spaces. Math Z 178:297–328 Giga Y (1985) Domain of fractional powers of the Stokes operator in L r spaces. Arch Rational Mech Anal 89:251–265 Giga Y, Miyakawa T (1985) Solutions in L r of the Navier–Stokes initial value problem. Arch Rational Mech Anal 89:267–281 Gunzburger M, Hou L, Sbovodny T (1991) Analysis and finite element approximation of optimal control problems for stationary Navier–Stokes equations with distributed and Neumann controls. Math Comp 57:123-151 Hartmann J (1937) Hg-dynamics I, Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Det Kgl Danske Videnskabernes Selskab Mat-fys Meddelelser XV 6:1–28. Hartmann J, Lazarus F (1937) Hg-dynamics II, Experimental investigations on the flow of mercury in a homogeneous magnetic field. Det Kgl Danske Videnskabernes Selskab Mat-fys Meddelelser XV 7:1–45 Henoch C, Stace J (1995) Experimental investigation of a salt water turbulent boundary layer modified by an applied streamwise magnetohydrodynamics body force. Phys Fluids 7:1371–1382 Henry D (1981) Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin Hille E, Phillips RS (1957) Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI Li X, Yao Y (1985) Maximum principle of distributed parameter systems with time lags. Lecture Notes in Control and Information Sciences, vol 75, pp 410–427. Springer-Verlag, Berlin Li X, Yong J (1995) Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston Masuda K (1975) On the stability of incompressible viscous fluid motions past objects. J Math Soc Japan 27:294–327

192


36.

Miyakawa T (1981) On the initial value problem for the Navier–Stokes equation in L r spaces. Hiroshima Math J 11:9–20 Moffatt K (1967) On the supression of turbulence by a uniform magnetic field. J Fluid Mech 28:571–592 Nosenchuk DM, Brown GL (1993) Discrete spatial control of wall shear stress in a turbulent boundary layer. Proc Internat Conf on Near Wall Turbulence Reed CB, Lykoudis PS (1978) The effect of a transverse magnetic field on shear turbulence. J Fluid Mech 89:147–171 Riesz F, Nagy BSz (1955) Functional Analysis. Ungar, New York Sobolevski PE (1964) Study of the Navier–Stokes equations by the methods of the theory of parabolic equations in Banach spaces. (Russian) Dokl Akad Nauk SSSR 156:745–748. (English) Soviet Math Dokl 7:720–723 Solonnikov VA (1977) Estimates for solutions of nonstationary Navier–Stokes equations. J Soviet Math 8:467–529. Sritharan SS (1992) An optimal control problem in exterior hydrodynamics. Proc Roy Soc Edinburgh Sect A 121:5–33 Sritharan SS (1993) Optimal feedback control of hydrodynamics: a progress report. Flow Control (M Gunzburger, ed). Springer-Verlag, New York Triebel H (1978) Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam Von Wahl W (1980) Regularity questions for the Navier–Stokes equations. Approximation Methods for Navier–Stokes Problems (R Rautmann, ed), pp 538–542. Lecture Notes in Mathematics, vol 771. Springer-Verlag, Berlin Von Wahl W (1985) The Equations of Navier–Stokes and Abstract Parabolic Equations. Vieweg, Braunschsweig/Wiesbaden Yosida K (1978) Functional Analysis. Springer-Verlag, Berlin

37. 38. 39. 40. 41.

42. 43. 44. 45. 46.

47. 48.

Accepted 3 December 1996

Optimal Control Problems with State Constraints in ... - Springer Link

Optimal Control Problems with State Constraints in ... - Springer Link

Suggest Documents

OPTIMAL CONTROL PROBLEMS WITH MIXED CONSTRAINTS 1

OPTIMAL CONTROL PROBLEMS WITH MIXED CONSTRAINTS 1

Introducing State Constraints in Optimal Control for Health Problems

Control Problems with State Constraints for the

Solving Optimal Control Problems With State

Analytic Solutions to Optimal Control Problems with Constraints

Solving nonlinear optimal control problems with state and control ...

Nearest feasible paths in optimal control problems - Springer Link

Optimal Control Problems with State Specific Jumps in ... - Google Sites

Optimal Control with State Constraints for Stochastic Evolution ...

Optimal attitude control with state constraints - Semantic Scholar

The Maximum Principle for Optimal Control Problems ... - Springer Link

A-Posteriori Error Estimates for Optimal Control Problems with State ...

Solving Optimal Control Problems With State ... - Semantic Scholar

Minimal Actuator Placement with Optimal Control Constraints

Control problems in quantum systems - Springer Link

Control problems in quantum systems - Springer Link

On Optimal Control Problems with Discontinuities - Core

Markov control processes with pathwise constraints - Springer Link

Handling constraints in optimal control with ... - Semantic Scholar

Superconvergence for Optimal Control Problems

OPTIMAL CONTROL OF PROBLEMS GOVERNED

Time optimal control computation with application to ... - Springer Link

Role of control constraints in quantum optimal control