LOCAL MINIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEMS SUBJECT TO INDEX ONE DIFFERENTIAL-ALGEBRAIC EQUATIONS MATTHIAS GERDTS∗ Abstract. Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-1 differential-algebraic equations, pure state constraints and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which frequently arise in practical applications. The local minimum principle is based on necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz like constraint qualification for the optimal control problem ensures the regularity of a local minimum. Key words. optimal control, necessary conditions, local minimum principle, index one differentialalgebraic equation system, state constraints AMS subject classifications. 49K15, 34A09
1. Introduction. Optimal control problems subject to ordinary differential equations have a wide range of applications in different disciplines like engineering sciences, chemical engineering, and economics. Necessary conditions known as ‘Maximum principles’ or ‘Minimum principles’ have been investigated intensively since the 1950’s. Early proofs of the maximum principle are given by Pontryagin et al. [28] and Hestenes [9]. Necessary conditions with pure state constraints are discussed in, e.g., Jacobsen et al. [12], Girsanov [6], Knobloch [14], Maurer [22, 23], Ioffe and Tihomirov [11], and Kreindler [15]. Neustadt [27] and Zeidan [31] discuss optimal control problems with mixed control-state constraints. Hartl et al. [7] provide a survey on maximum principles for optimal control problems with state constraints including an extensive list of references. Necessary conditions for variational problems, i.e. smooth optimal control problems, are developed in Bryson and Ho [1]. Second order necessary conditions and sufficient conditions are stated in Zeidan[31]. Sufficient conditions are also presented in Maurer [24] and Malanowski [21]. Necessary conditions are not only interesting from a theoretical point of view, but also provide the basis of the so-called indirect approach for solving optimal control problems numerically. In this approach the minimum principle is exploited and usually leads to a multi-point boundary value problem, which is solved numerically by, e.g., the multiple shooting method. Nevertheless, even for the direct approach, which is based on a suitable discretization of the optimal control problem, the minimum principle is very important for the post-optimal approximation of adjoints. In this context, the multipliers resulting from the formulation of the necessary Fritz-John conditions for the finite dimensional discretized optimal control problem have to be related to the multipliers of the original infinite dimensional optimal control problem in an appropriate way. It is evident that this requires the knowledge of necessary conditions for the optimal control problem. We will extend the results for optimal control problems with ordinary differential equations to problems involving differential-algebraic equations (DAE’s). Differentialalgebraic equations are composite systems of differential equations and algebraic equa∗ Fachbereich Mathematik (OA), Universit¨ at Hamburg, Bundesstraße 55, 20146 Hamburg, Germany (
[email protected]).
1
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M. GERDTS
tions. Particularly, we will discuss DAE systems of type x(t) ˙ = f (t, x(t), y(t), u(t)) 0ny = g(t, x(t), y(t), u(t))
a.e. in [t0 , tf ], a.e. in [t0 , tf ].
(1.1) (1.2)
Herein, x(t) ∈ Rnx is referred to as differential variable and y(t) ∈ Rny as algebraic variable. Correspondingly, (1.1) is called differential equation and (1.2) algebraic equation. The variable u is a control variable, which is an external function and allows to control the system in an appropriate way. In this paper we will restrict the discussion to so-called semi-explicit index-1 DAE systems. Semi-explicit index-1 DAE systems are characterized by the postulation that the matrix gy0 (t, x(t), y(t), u(t)) is non-singular. Semi-explicit Index-1 systems often occur in process system engineering, cf. Hinsberger [10], but also in vehicle simulation, cf. Gerdts [5], and many other fields of applications. Necessary conditions for optimal control problems subject to index-1 DAE systems without state constraints and without mixed control-state constraints can be found in de Pinho and Vinter [2]. Implicit control systems are discussed in Devdariani and Ledyaev [3]. In our representation, we followed to a large extend the approaches in Kirsch et al. [13] and Machielsen [20]. We investigate nonlinear optimal control problems of the subsequent form. Let [t0 , tf ] ⊂ R be a non-empty and bounded interval with fixed time points t0 , tf and U ⊆ Rnu a closed and convex set with non-empty interior. Let ϕ : Rnx × Rnx → R, f0 : [t0 , tf ] × Rnx × Rny × Rnu f : [t0 , tf ] × Rnx × Rny × Rnu g : [t0 , tf ] × Rnx × Rny × Rnu ψ : R n x × R nx → R n ψ , c : [t0 , tf ] × Rnx × Rny × Rnu s : [t0 , tf ] × Rnx → Rns
→ R, → Rnx , → Rny , → Rnc ,
be mappings. For n ∈ N, the space L∞ ([t0 , tf ], Rn ) consists of all measurable functions h : [t0 , tf ] → Rn with khk∞ := ess supkh(t)k < ∞, t0 ≤t≤tf
where k · k denotes the Euclidian norm on Rn . With this norm, L∞ ([t0 , tf ], Rn ) becomes a Banach space. The space W 1,∞ ([t0 , tf ], Rn ) consists of all absolutely continuous functions h : [t0 , tf ] → Rn with khk1,∞ := max{khk∞ , kh0 k∞ } < ∞, where h0 denotes the first derivative of h. The space W 1,∞ ([a, b], Rn ) endowed with the norm k · k1,∞ is a Banach space. We will denote the null element of a general Banach space X by ΘX or, if no confusion is possible, simply by Θ. We will use 0n if X = Rn and 0 if X = R. We consider Problem 1.1 (DAE optimal control problem). Find an absolutely continuous differential variable x ∈ W 1,∞ ([t0 , tf ], Rnx ), an essentially bounded algebraic variable y ∈ L∞ ([t0 , tf ], Rny ), and an essentially bounded
LOCAL MINIMUM PRINCIPLE
control variable u ∈ L∞ ([t0 , tf ], Rnu ) such that the objective function Z tf ϕ(x(t0 ), x(tf )) + f0 (t, x(t), y(t), u(t))dt
3
(1.3)
t0
is minimized subject to the semi-explicit differential algebraic equation (1.1)-(1.2), the boundary conditions ψ(x(t0 ), x(tf )) = 0nψ ,
(1.4)
the mixed control-state constraints c(t, x(t), y(t), u(t)) ≤ 0nc
a.e. in [t0 , tf ],
(1.5)
the pure state constraints s(t, x(t)) ≤ 0ns
in [t0 , tf ],
(1.6)
and the set constraints u(t) ∈ U
a.e. in [t0 , tf ].
(1.7)
Some definitions and terminologies are in order. (x, y, u) ∈ W 1,∞ ([t0 , tf ], Rnx ) × L ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu ) is called admissible or feasible for the optimal control problem 1.1, if the constraints (1.1)-(1.7) are fulfilled. An admissible pair ∞
(ˆ x, yˆ, u ˆ) ∈ W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu ) is called a weak local minimum of Problem 1.1, if there exists ε > 0 such that Z tf ϕ(ˆ x(t0 ), x ˆ(tf )) + f0 (t, x ˆ(t), yˆ(t), u ˆ(t))dt t0 tf
Z ≤ ϕ(x(t0 ), x(tf )) +
f0 (t, x(t), y(t), u(t))dt t0
holds for all admissible (x, y, u) with kx− x ˆk1,∞ < ε, ky − yˆk∞ < ε, and ku− u ˆk∞ < ε. An admissible pair (ˆ x, yˆ, u ˆ) ∈ W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu ) is called a strong local minimum of Problem 1.1, if there exists ε > 0 such that Z tf ϕ(ˆ x(t0 ), x ˆ(tf )) + f0 (t, x ˆ(t), yˆ(t), u ˆ(t))dt t0 tf
Z ≤ ϕ(x(t0 ), x(tf )) +
f0 (t, x(t), y(t), u(t))dt t0
holds for all admissible (x, y, u) with kx − x ˆk∞ < ε. Notice, that strong local minima are also weak local minima. The converse is not true. Strong local minima are minimal w.r.t. a larger class of algebraic variables and controls. Weak local minima are only optimal w.r.t. all algebraic variables and controls in a L∞ -neighborhood.
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The paper is organized as follows. In Section 2 necessary conditions for general infinite optimization problems known from the literature are summarized. These necessary conditions are formulated for the optimal control problem in Section 3. Explicit representations of the multipliers involved in the necessary conditions are derived in Section 4. These explicit representations are exploited in order to state the main result of the paper in Section 5 – the local minimum principle for optimal control problems of type (1.1). In addition, some important special cases are discussed. Section 6 summarizes a constraint qualification which ensures the regularity of a local minimum. Finally, some concluding remarks close the paper. 2. Necessary conditions for infinite optimization problems. We summarize results for optimization problems in Banach spaces. Let (X, k·kX ), (Y, k·kY ), (Z, k· kZ ) be Banach spaces, f : X → R a functional and g : X → Y , h : X → Z operators. The topological dual spaces of X, Y, Z are denoted by X ∗ , Y ∗ , Z ∗ , respectively. Let S ⊆ X be a closed convex set with non-empty interior and K ⊆ Y a closed convex cone with vertex at zero. K + := {y ∗ ∈ Y ∗ | y ∗ (y) ≥ 0 for all y ∈ K} is referred to as the positive dual cone of K. Consider the optimization problem min f (x) s.t. x ∈ S, g(x) ∈ K, h(x) = ΘZ .
(2.1)
Theorem 2.1. Let x ˆ be a local minimum of the optimization problem (2.1). Let f : X → R and g : X → Y be Fr´echet-differentiable at x ˆ, h : X → Z continuously Fr´echet-differentiable at x ˆ, K ⊆ Y a closed convex cone with vertex at zero, and S a closed convex set with non-empty interior. Furthermore, assume that (i) int(K) 6= ∅ and (ii) im(h0 (ˆ x)) is not a proper dense subspace in Z. Then there exist nontrivial multipliers (0, ΘY ∗ , ΘZ ∗ ) 6= (l0 , η ∗ , λ∗ ) ∈ R × Y ∗ × Z ∗ such that l0 ≥ 0, η∗ ∈ K + , ∗ η (g(ˆ x)) = 0, 0 ∗ 0 ∗ 0 l0 f (ˆ x)(d) − η (g (ˆ x)(d)) − λ (h (ˆ x)(d)) ≥ 0, ∀d ∈ S − {ˆ x}.
(2.2) (2.3) (2.4) (2.5)
Proof. Kurcyusz [16], Th. 3.1, Cor. 4.2, Lempio [17] Conditions which ensure that the multiplier l0 in Theorem 2.1 is not zero are called regularity conditions or constraint qualifications. In this case, without loss of generality l0 can be normalized to one. The subsequent Mangasarian-Fromowitz like condition is such a regularity condition. Corollary 2.2. Let the assumptions of Theorem 2.1 be satisfied. In addition, let the following assumptions be fulfilled: (i) h0 (ˆ x) is surjective. (ii) There exists some dˆ ∈ int(S − {ˆ x}) with ˆ = ΘZ , h0 (ˆ x)(d) 0 ˆ ∈ int(K − {g(ˆ g (ˆ x)(d) x)}).
(2.6) (2.7)
LOCAL MINIMUM PRINCIPLE
5
Then, the assertions of Theorem 2.1 hold with l0 = 1. Proof. Follows from Theorem 2.1 by contradiction, if it is assumed, that the assertions in Theorem 2.1 hold with l0 = 0. Remark 2.3. The above regularity condition of Mangasarian-Fromowitz is equivalent to the famous regularity condition of Robinson [29] (ΘY , ΘZ ) ∈ int{(g 0 (ˆ x)(s − x ˆ) − k + g(ˆ x), h0 (ˆ x)(s − x ˆ)) | s ∈ S, k ∈ K}.
(2.8)
Robinson [29] postulated this condition in the context of stability analysis of generalized inequalities. Zowe and Kurcyusz [32] used it to show l0 = 1. 3. Abstract formulation of the optimal control problem. In order to obtain necessary conditions for a weak local minimum, we reformulate the general problem 1.1 as an optimization problem in appropriate Banach spaces. We follow the approaches of Kirsch et al. [13] and Machielsen [20]. For notational convenience throughout this chapter we will use the abbreviations ϕ0x0 := ϕ0x0 (ˆ x(t0 ), x ˆ(tf )), 0 0 fx [t] := fx (t, x ˆ(t), yˆ(t), u ˆ(t)), 0 0 0 and in a similar way ϕ0xf , f0,x [t], f0,y [t], f0,u [t], c0x [t], c0y [t], c0u [t], s0x [t], fy0 [t], 0 0 0 0 0 0 fu [t], gx [t], gy [t], gu [t], ψx0 , ψxf for the respective derivatives. The variables (x, y, u) are taken as elements from the Banach space
X := W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × L∞ ([t0 , tf ], Rnu ) endowed with the norm k(x, y, u)kX := max{kxk1,∞ , kyk∞ , kuk∞ }. The objective function F : X → R is given by Z tf F (x, y, u) := ϕ(x(t0 ), x(tf )) + f0 (t, x(t), y(t), u(t))dt. t0
If ϕ and f0 are continuous w.r.t. all arguments and continuously differentiable w.r.t. to x, y, and u, then F is Fr´echet-differentiable at (ˆ x, yˆ, u ˆ) with F 0 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = ϕ0x0 δx(t0 ) + ϕ0xf δx(tf ) Z tf 0 0 0 + f0,x [t]δx(t) + f0,y [t]δy(t) + f0,u [t]δu(t)dt, t0
cf. Kirsch et al. [13], p. 94. Now we collect the equality constraints. The space Z := L∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × Rnψ endowed with the norm k(z1 , z2 , z3 )kZ := max{kz1 k∞ , kz2 k∞ , kz3 k2 } is a Banach space. The equality constraints of the optimal control problem are given by H(x, y, u) = ΘZ ,
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M. GERDTS
where H = (H1 , H2 , H3 ) : X → Z with H1 (x, y, u) = f (·, x(·), y(·), u(·)) − x(·), ˙ H2 (x, y, u) = g(·, x(·), y(·), u(·)), H3 (x, y, u) = −ψ(x(t0 ), x(tf )). If f , g, and ψ are continuous w.r.t. all arguments and continuously differentiable w.r.t. to x, y, and u, then H is continuously Fr´echet-differentiable at (ˆ x, yˆ, u ˆ) with 0 H1 (ˆ x, yˆ, u ˆ)(δx, δy, δu) x, yˆ, u ˆ)(δx, δy, δu) H 0 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = H20 (ˆ 0 H3 (ˆ x, yˆ, u ˆ)(δx, δy, δu) and ˙ H10 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = fx0 [·]δx(·) + fy0 [·]δy(·) + fu0 [·]δu(·) − δx(·), 0 0 0 0 H2 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = gx [·]δx(·) + gy [·]δy(·) + gu [·]δu(·), 0 H3 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = −ψx0 0 δx(t0 ) − ψx0 f δx(tf ), cf. Kirsch et al. [13], p. 95. Now we collect the inequality constraints. The space Y := L∞ ([t0 , tf ], Rnc ) × C([t0 , tf ], Rns ) endowed with the norm k(y1 , y2 )kY := max{ky1 k∞ , ky2 k∞ } is a Banach space. The inequality constraints of the optimal control problem are given by G(x, y, u) ∈ K, where G = (G1 , G2 ) : X → Y and G1 (x, y, u) = −c(·, x(·), y(·), u(·)), G2 (x, y, u) = −s(·, x(·)). The cone K := K1 × K2 ⊆ Y is defined by K1 := {z ∈ L∞ ([t0 , tf ], Rnc ) | z(t) ≥ 0nc a.e. in [t0 , tf ]}, K2 := {z ∈ C([t0 , tf ], Rns ) | z(t) ≥ 0ns in [t0 , tf ]}. If c and s are continuous w.r.t. all arguments and continuously differentiable w.r.t. to x, y, u, then G is continuously Fr´echet-differentiable at (ˆ x, yˆ, u ˆ) with 0 G1 (ˆ x, yˆ, u ˆ)(δx, δy, δu) G0 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = G02 (ˆ x, yˆ, u ˆ)(δx, δy, δu) and G01 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = −c0x [·]δx(·) − c0y [·]δy(·) − c0u [·]δu(·), G02 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = −s0x [·]δx(·).
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LOCAL MINIMUM PRINCIPLE
Finally, the set S ⊆ X is given by S := W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × Uad , where Uad := {u ∈ L∞ ([t0 , tf ], Rnu ) | u(t) ∈ U a.e. in [t0 , tf ]}. Summarizing, the optimal control problem 1.1 is equivalent with Problem 3.1. Find (x, y, u) ∈ X such that F (x, y, u) is minimized subject to the constraints G(x, y, u) ∈ K,
H(x, y, u) = ΘZ ,
(x, y, u) ∈ S.
We intend to apply the necessary conditions in Theorem 2.1 to Problem 3.1. In order to show the surjectivity of the equality constraints, we need the following results about linear differential algebraic equations and boundary value problems. Lemma 3.2. Consider the linear DAE x(t) ˙ = A1 (t)x(t) + B1 (t)y(t) + h1 (t), 0ny = A2 (t)x(t) + B2 (t)y(t) + h2 (t),
(3.1) (3.2)
where A1 (t) ∈ Rnx ×nx , A2 (t) ∈ Rny ×nx , B1 (t) ∈ Rnx ×ny , B2 (t) ∈ Rny ×ny are time dependent matrix functions with entries from L∞ ([t0 , tf ], R) and h1 ∈ L∞ ([t0 , tf ], Rnx ), h2 ∈ L∞ ([t0 , tf ], Rny ). Let B2 (t) be non-singular almost everywhere in [t0 , tf ] and let B2 (t)−1 be essentially bounded. Define A(t) := A1 (t) − B1 (t)B2 (t)−1 A2 (t), h(t) := h1 (t) − B1 (t)B2 (t)−1 h2 (t). (a) The initial value problem given by (3.1)-(3.2) together with the initial value x(t0 ) = x0 has a unique solution x ∈ W 1,∞ ([t0 , tf ], Rnx ), y ∈ L∞ ([t0 , tf ], Rny ) for every x0 ∈ Rnx , every h1 ∈ L∞ ([t0 , tf ], Rnx ), and every h2 ∈ L∞ ([t0 , tf ], Rny ). The solution is given by Z t −1 x(t) = Φ(t) x0 + Φ (τ )h(τ )dτ , in [t0 , tf ], (3.3) t0
y(t) = −B2 (t)−1 (h2 (t) + A2 (t)x(t))
a.e. in [t0 , tf ],
(3.4)
where the fundamental system Φ(t) ∈ Rnx ×nx is the unique solution of ˙ Φ(t) = A(t)Φ(t),
Φ(t0 ) = Inx .
(3.5)
(b) Let a vector b ∈ Rr and matrices C0 , Cf ∈ Rr×nx be given, such that rank (C0 Φ(t0 ) + Cf Φ(tf )) = r holds for the fundamental solution Φ from (a). Then, the boundary value problem given by (3.1)-(3.2) together with the boundary condition C0 x(t0 ) + Cf x(tf ) = b has a solution for every b ∈ Rr .
(3.6)
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M. GERDTS
Proof. see Appendix A Part (a) of Lemma 3.2 enables us to formulate necessary conditions for the optimal control problem 1.1 if we impose Assumption 3.3. Let the index of the DAE (1.1)-(1.2) be one, i.e. let gy0 [t] := gy0 (t, x ˆ(t), yˆ(t), u ˆ(t)) be non-singular a.e. in [t0 , tf ] and let gy0 [t]−1 be essentially bounded. Remark 3.4. Notice, that by Assumption 3.3 the algebraic equation (1.2) can be solved for yˆ and yˆ can be expressed as a function of t, x ˆ, u ˆ. By exploitation of the minimum principle known for ordinary differential equations this approach would lead to a second version of the proof of the upcoming minimum principle for DAE systems. However, we do not follow this approach, since we think it is important to give a direct proof in order to gain more insights in the underlying functional analytic tools. Moreover, in the future more general DAE systems will be considered, where this simple transformation is not possible anymore. Furthermore, we need the following abstract result. Lemma 3.5. Let X, Y be Banach spaces. Let T : X → Y × Rn be defined by T (x) = (T1 (x), T1 (x)), where T1 : X → Y is a linear, continuous, and surjective operator and T2 : X → Rn is linear and continuous. Then im(T ) = T (X) is closed in Y × Rn . Proof. see Appendix A Theorem 3.6 (Necessary Conditions). Let the following assumptions be fulfilled for the optimal control problem 1.1. (i) Let the functions ϕ, f0 , f, g, ψ, c, s be continuous w.r.t. all arguments and continuously differentiable w.r.t. x, y, and u. (ii) Let U ⊆ Rnu be a closed and convex set with non-empty interior. (iii) Let (ˆ x, yˆ, u ˆ) ∈ X be a weak local minimum of the optimal control problem. (iv) Let Assumption 3.3 be valid. Then there exist non-trivial multipliers l0 ∈ R, η ∗ ∈ Y ∗ , λ∗ ∈ Z ∗ with l0 ≥ 0, η∗ ∈ K + , ∗ η (G(ˆ x, yˆ, u ˆ)) = 0, 0 l0 F (ˆ x, yˆ, u ˆ)(x − x ˆ, y − yˆ, u − u ˆ) ∗ 0 −η (G (ˆ x, yˆ, u ˆ)(x − x ˆ, y − yˆ, u − u ˆ)) ∗ 0 −λ (H (ˆ x, yˆ, u ˆ)(x − x ˆ, y − yˆ, u − u ˆ)) ≥ 0 ∀(x, y, u) ∈ S.
(3.7) (3.8) (3.9)
(3.10)
Proof. We show, that all assumptions of Theorem 2.1 are satisfied. Observe, that K is a closed convex cone with vertex at zero, int(K) is non-empty, the functions F and G are Fr´echet-differentiable, and H is continuously Fr´echet-differentiable due to the smoothness assumptions. Since U is supposed to be closed and convex with nonempty interior, the set S is closed and convex with non-empty interior. It remains to show that im(H 0 (ˆ x, yˆ, u ˆ)) is not a proper dense subset of Z. According to part (a) of x, yˆ, u ˆ)) is continuous, linear and surjective. Lemma 3.2 the operator (H10 (ˆ x, yˆ, u ˆ), H20 (ˆ Thus, we can apply Lemma 3.5, which yields that the image of H 0 (ˆ x, yˆ, u ˆ) is closed in Z and hence im(H 0 (ˆ x, yˆ, u ˆ)) is not a proper dense subset in Z. Hence, all assumptions of Theorem 2.1 are satisfied and Theorem 2.1 yields (3.7)-(3.10).
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LOCAL MINIMUM PRINCIPLE
Notice, that the multipliers are elements of the following dual spaces: ∗
∗
η ∗ := (η1∗ , η2∗ ) ∈ Y ∗ = (L∞ ([t0 , tf ], Rnc )) × (C([t0 , tf ], Rns )) , ∗ ∗ λ∗ := (λ∗1 , λ∗2 , σ) ∈ Z ∗ = (L∞ ([t0 , tf ], Rnx )) × (L∞ ([t0 , tf ], Rny )) × Rnψ . In the sequel, the special structure of the functions F , G, and H in (3.10) is exploited. Furthermore, the fact, that the variational inequality (3.10) holds for all (x, y, u) ∈ S, i.e. for all x ∈ W 1,∞ ([t0 , tf ], Rnx ), all y ∈ L∞ ([t0 , tf ], Rny ), and all u ∈ Uad , is used to derive three single variational equalities and inequalities, respectively. Evaluation of the Fr´echet-derivatives in (3.10) and setting y = yˆ, u = u ˆ yields the variational equality l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) +
Z
tf
0 l0 f0,x [t]δx(t)dt
t0
˙ +η1∗ (c0x [·]δx(·)) + η2∗ (s0x [·]δx(·)) + λ∗1 (δx(·) − fx0 [·]δx(·)) − λ∗2 (gx0 [·]δx(·))
= 0, (3.11)
which holds for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). Similarly, we find Z
tf
0 [t]δy(t)dt + η1∗ (c0y [·]δy(·)) − λ∗1 (fy0 [·]δy(·)) − λ∗2 (gy0 [·]δy(·)) l0 f0,y
= 0
(3.12)
t0
for all δy ∈ L∞ ([t0 , tf ], Rny ) and Z
tf
0 l0 f0,u [t]δu(t)dt + η1∗ (c0u [·]δu(·)) − λ∗1 (fu0 [·]δu(·)) − λ∗2 (gu0 [·]δu(·)) ≥ 0
t0
(3.13) for all δu ∈ Uad − {ˆ u}. Recall, that the multiplier η2∗ is an element of the dual space of the space of continuous functions. Hence, by Riesz’ representation theorem the functional η2∗ admits the representation η2∗ (h)
=
ns Z X i=1
tf
hi (t)dµi (t)
(3.14)
t0
for every continuous function h ∈ C([t0 , tf ], Rns ). Herein, µi , i = 1, . . . , ns are functions of bounded variation. To make the representation unique, we choose µi , i = 1, . . . , ns from the space N BV ([t0 , tf ], R), i.e. the space of normalized functions of bounded variation which are continuous from the right in (t0 , tf ) and satisfy µi (t0 ) = 0 for i = 1, . . . , ns , cf. Luenberger [19] and Natanson [26]. Replacing η2∗ in (3.11) by (3.14) and rearranging terms we obtain Z tf 0 l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) + l0 f0,x [t]δx(t)dt t0 Z n s tf X + s0i,x [t]δx(t)dµi (t) + η1∗ (c0x [·]δx(·)) i=1
t0
˙ +λ∗1 (δx(·) − fx0 [·]δx(·)) − λ∗2 (gx0 [·]δx(·))
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ).
= 0 (3.15)
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M. GERDTS
4. Representation of Multipliers. In this section, useful representations of the functionals λ∗1 λ∗2 , and η1∗ are derived and properties of Stieltjes integrals will be exploited extensively. These results about Stieltjes integrals can be found in the books of Natanson [26] and Widder [30]. According to Lemma 3.2 the initial value problem ˙ δx(t) = fx0 [t]δx(t) + fy0 [t]δy(t) + h1 (t), 0ny = gx0 [t]δx(t) + gy0 [t]δy(t) + h2 (t),
δx(t0 ) = 0nx ,
(4.1) (4.2)
has a solution for every h1 ∈ L∞ ([t0 , tf ], Rnx ) and every h2 ∈ L∞ ([t0 , tf ], Rny ). According to (3.3) and (3.4) in Lemma 3.2 the solution is given by Z t Φ−1 (τ )h(τ )dτ, (4.3) δx(t) = Φ(t) t0 −1 0 gy [t]
(gx0 [t]δx(t) + h2 (t)) Z t −1 0 Φ−1 (τ )h(τ )dτ + h2 (t) , gx [t]Φ(t) = − gy0 [t]
δy(t) = −
(4.4)
t0
where −1 h(t) = h1 (t) − fy0 [t] gy0 [t] h2 (t) ˙ and Φ is the solution of Φ(t) = A(t)Φ(t), Φ(t0 ) = Inx with −1 0 A(t) = fx0 [t] − fy0 [t] gy0 [t] gx [t]. Now, let h1 ∈ L∞ ([t0 , tf ], Rnx ) and h2 ∈ L∞ ([t0 , tf ], Rny ) be arbitrary. Adding equations (3.12) and (3.15) and exploiting the linearity of the functionals leads to l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) Z tf ns Z tf X 0 0 (4.5) + l0 f0,x [t]δx(t) + l0 f0,y [t]δy(t)dt + s0i,x [t]δx(t)dµi (t) t0
i=1
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0.
Introducing the solution formulas into equation (4.5) and combining terms yields Z tf 0 > 0 l0 ϕxf + σ ψxf Φ(tf ) Φ−1 (t)h1 (t)dt t 0 Z tf −1 − l0 ϕ0xf + σ > ψx0 f Φ(tf ) Φ−1 (t)fy0 [t] gy0 [t] h2 (t)dt t0 Z t Z tf + l0 fˆ0 [t]Φ(t) Φ−1 (τ )h1 (τ )dτ dt t0 t0 Z Z tf t −1 −1 0 0 ˆ − l0 f0 [t]Φ(t) Φ (τ )fy [τ ] gy [τ ] h2 (τ )dτ dt t0 t0 (4.6) Z tf −1 0 0 − l0 f0,y [t] gy [t] h2 (t)dt t0 Z Z ns tf t X + s0i,x [t]Φ(t) Φ−1 (τ )h1 (τ )dτ dµi (t) −
ns Z X i=1
i=1 tf
t0
s0i,x [t]Φ(t)
t0
t0
Z
t
−1 Φ−1 (τ )fy0 [τ ] gy0 [τ ] h2 (τ )dτ
dµi (t)
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0,
11
LOCAL MINIMUM PRINCIPLE
where −1 0 0 0 fˆ0 [t] := f0,x [t] − f0,y [t] gy0 [t] gx [t]. Integration by parts yields Z t Z tf Z Φ−1 (τ )h1 (τ )dτ dt = fˆ0 [t]Φ(t) t0
t0
tf
tf
Z
t0
fˆ0 [τ ]Φ(τ )dτ
Φ−1 (t)h1 (t)dt
t
and tf
Z
fˆ0 [t]Φ(t)
t0
Z
t
Z
−1 Φ−1 (τ )fy0 [τ ] gy0 [τ ] h2 (τ )dτ
t0 tf Z tf
=
fˆ0 [τ ]Φ(τ )dτ
dt
−1 h2 (t)dt. Φ−1 (t)fy0 [t] gy0 [t]
t
t0
Introducing these relations into (4.6) leads to Z tf Z tf 0 > 0 ˆ l0 ϕxf + σ ψxf Φ(tf ) + l0 f0 [τ ]Φ(τ )dτ Φ−1 (t)h1 (t)dt t0 t Z tf l0 ϕ0xf + σ > ψx0 f Φ(tf ) − t0 Z tf −1 ˆ l0 f0 [τ ]Φ(τ )dτ Φ−1 (t)fy0 [t] gy0 [t] + h2 (t)dt t Z tf −1 0 − l0 f0,y [t] gy0 [t] h2 (t)dt t0 Z Z n s tf t X + s0i,x [t]Φ(t) Φ−1 (τ )h1 (τ )dτ dµi (t) −
ns Z X i=1
t0
i=1 tf
t0 t
Z
s0i,x [t]Φ(t)
Φ
t0
(4.7)
−1
(τ )fy0 [τ ]
−1 gy0 [τ ]
h2 (τ )dτ
dµi (t)
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0,
The Riemann-Stieltjes integrals are to be transformed using integration by parts. Therefore, let a : [t0 , tf ] → Rn be continuous, b : [t0 , tf ] → Rn absolutely continuous and µ : [t0 , tf ] → R of bounded variation. Then, using integration by parts for Riemann-Stieltjes integrals leads to Z tf n Z tf X a(t)> b(t)dµ(t) = ai (t)bi (t)dµ(t) t0
t0
i=1
=
=
n Z X
tf
Z bi (t)d
t0
i=1 n X
t
ai (τ )dµ(τ )
t0
Z
t
tf ai (τ )dµ(τ ) · bi (t)
t0
i=1
t0
Z
tf
Z
! ai (τ )dµ(τ ) dbi (t)
t
− t0
Z
t
= t0
Z
tf
= t0
t0
tf Z − a(τ )> dµ(τ ) · b(t)
tf
Z
t0
t0
Z > a(τ ) dµ(τ ) · b(tf ) −
tf
t0
t
a(τ )> dµ(τ ) b0 (t)dt
t0
Z
t
t0
a(τ ) dµ(τ ) b0 (t)dt. >
12
M. GERDTS
Application of this formula to (4.7) where a(t)> = s0i,x [t]Φ(t) and
t
Z
t
Z
Φ−1 (τ )h1 (τ )dτ
b(t) =
resp.
b(t) =
t0
−1 Φ−1 (τ )fy0 [τ ] gy0 [τ ] h2 (τ )dτ
t0
yields
Z
tf
t
Z
s0i,x [t]Φ(t)
Φ
−1
(τ )h1 (τ )dτ
dµi (t)
t0
t0
Z
tf
Z
tf
s0i,x [τ ]Φ(τ )dµi (τ ) Φ−1 (t)h1 (t)dt.
= t0
t
and
Z
tf
s0i,x [t]Φ(t)
t0
Z
tf
t
Z
t0 tf
Z
= t0
−1 Φ−1 (τ )fy0 [τ ] gy0 [τ ] h2 (τ )dτ
dµi (t)
−1 s0i,x [τ ]Φ(τ )dµi (τ ) Φ−1 (t)fy0 [t] gy0 [t] h2 (t)dt.
t
Substitution into (4.7) yields
Z
tf
t0
l0 ϕ0xf
+σ
>
ψx0 f
Φ(tf ) + ns Z X
− t0
l0 ϕ0xf +
+σ
>
ns Z X i=1
ψx0 f
tf
! Φ−1 (t)h1 (t)dt
tf
s0i,x [τ ]Φ(τ )dµi (τ )
t
i=1 tf
l0 fˆ0 [τ ]Φ(τ )dτ
t
+ Z
tf
Z
Z Φ(tf ) + t
tf
l0 fˆ0 [τ ]Φ(τ )dτ !
s0i,x [τ ]Φ(τ )dµi (τ )
t
−1 Φ−1 (t)fy0 [t] gy0 [t] h2 (t)dt Z tf −1 0 − l0 fy,0 [t] gy0 [t] h2 (t)dt t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0. (4.8)
Equation (4.8) is equivalent with
Z
tf
>
Z
tf
p1 (t) h1 (t)dt + t0
p2 (t)> h2 (t)dt
(4.9)
t0
+η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) + λ∗1 (h1 (·)) + λ∗2 (h2 (·))
= 0,
13
LOCAL MINIMUM PRINCIPLE
where p1 (t)> :=
Z l0 ϕ0xf + σ > ψx0 f Φ(tf ) +
tf
l0 fˆ0 [τ ]Φ(τ )dτ
t
+
ns Z X
tf
! Φ−1 (t),
s0i,x [τ ]Φ(τ )dµi (τ )
(4.10)
t
i=1
−1 0 p2 (t)> := −l0 fy,0 [t] gy0 [t] Z 0 > 0 l0 ϕxf + σ ψxf Φ(tf ) + −
tf
l0 fˆ0 [τ ]Φ(τ )dτ
t
+
ns Z X i=1
tf
! −1 Φ−1 (t)fy0 [t] gy0 [t] . (4.11)
s0i,x [τ ]Φ(τ )dµi (τ )
t
(4.9) and (3.13) will be exploited in order to derive explicit representations of the functionals λ∗1 , λ∗2 , and η1∗ . This is possible if either there are no mixed controlstate constraints (1.5) present in the optimal control problem, or if there are no set constraints (1.7), i.e. U = Rnu . Corollary 4.1. Let the assumptions of Theorem 3.6 be valid and let there be no mixed control-state constraints (1.5) in the optimal control problem 1.1. Then there exist functions p1 ∈ BV ([t0 , tf ], Rnx ) and p2 ∈ L∞ ([t0 , tf ], Rny ) with λ∗1 (h1 (·))
Z
tf
=−
λ∗2 (h2 (·)) = −
t0 Z tf
p1 (t)> h1 (t)dt,
(4.12)
p2 (t)> h2 (t)dt
(4.13)
t0
for every h1 ∈ L∞ ([t0 , tf ], Rnx ) and every h2 ∈ L∞ ([t0 , tf ], Rny ) . p1 , p2 are given by (4.10)-(4.11). Proof. The assertions follow from (4.1), (4.2), (4.9), (4.10)-(4.11) and the preceeding considerations, if we choose h1 (·) = Θ respectively h2 (·) = Θ. p1 is of bounded variation, since the Riemann-Stieltjes integral in (4.10) is of bounded variation and Φ, Φ−1 , and the first integral are absolutely continuous. p2 is essentially bounded, since all terms in (4.11) are so. Notice, that p1 in (4.10) is even absolutely continuous, if no pure state constraints (1.6) are present in Problem 1.1. Corollary 4.2. Let the assumptions of Theorem 3.6 be valid and let U = Rnu . Let rank(c0u [t]) = nc
(4.14)
hold almost everywhere in [t0 , tf ]. Furthermore, let the matrix gy0 [t] − gu0 [t](c0u [t])+ c0y [t]
(4.15)
be non-singular almost everywhere in [t0 , tf ], where (c0u [t])+ := c0u [t]> c0u [t]c0u [t]>
−1
.
(4.16)
14
M. GERDTS
denotes the pseudo-inverse of c0u [t]. Then there exist functions λ1 ∈ BV ([t0 , tf ], Rnx ), λ2 ∈ L∞ ([t0 , tf ], Rny ), η ∈ ∞ L ([t0 , tf ], Rnc ) with Z tf ∗ λ1 (h1 (·)) = − λ1 (t)> h1 (t)dt, λ∗2 (h2 (·)) = −
t0 tf
Z
λ2 (t)> h2 (t)dt,
t0
η1∗ (k(·)) =
Z
tf
η(t)> k(t)dt
t0 ∞
for every h1 ∈ L ([t0 , tf ], R ), h2 ∈ L∞ ([t0 , tf ], Rny ), and k ∈ L∞ ([t0 , tf ], Rnc ). Proof. The assumption U = Rnu implies Uad = L∞ ([t0 , tf ], Rnu ) and hence, inequality (3.13) turns into the equality Z tf 0 l0 f0,u [t]δu(t)dt + η1∗ (c0u [·]δu(·)) − λ∗1 (fu0 [·]δu(·)) − λ∗2 (gu0 [·]δu(·)) = 0 (4.17) nx
t0
for all δu ∈ L∞ ([t0 , tf ], Rnu ). Equation (4.9) for the particular choices h1 (·) = fu0 [·]δu(·) and h2 (·) = gu0 [·]δu(·) yields Rt R tf p (t)> fu0 [t]δu(t)dt + t0f p2 (t)> gu0 [t]δu(t)dt + η1∗ (c0x [·]δx(·) + c0y [·]δy(·)) t0 1 = −λ∗1 (fu0 [·]δu(·)) − λ∗2 (gu0 [·]δu(·)), (4.18) where δx and δy are determined by ˙ δx(t) = fx0 [t]δx(t) + fy0 [t]δy(t) + fu0 [t]δu(t), 0ny = gx0 [t]δx(t) + gy0 [t]δy(t) + gu0 [t]δu(t),
δx(t0 ) = 0nx ,
(4.19) (4.20)
The latter follows from (4.1), (4.2), and Lemma 3.2. Introducing (4.18) into (4.17) and exploiting the linearity of the functional η1∗ yields Z tf Hu0 [t]δu(t)dt + η1∗ (k(·)) = 0 (4.21) t0
with 0 Hu0 [t] := l0 f0,u [t] + p1 (t)> fu0 [t] + p2 (t)> gu0 [t]
and k(t) := c0x [t]δx(t) + c0y [·]δy(·) + c0u [t]δu(t).
(4.22)
Due to the rank assumption (4.14) equation (4.22) can be solved for δu with δu(t) = (c0u [t])+ k(t) − c0x [t]δx(t) − c0y [t]δy(t) , (4.23) where (c0u [t])+ denotes the pseudo-inverse of c0u [t]. Using the relation (4.23), equation (4.21) becomes Z tf (4.24) Hu0 [t](c0u [t])+ k(t) − c0x [t]δx(t) − c0y [t]δy(t) dt + η1∗ (k(·)) = 0. t0
15
LOCAL MINIMUM PRINCIPLE
Replacing δu in (4.19) and (4.20) by (4.23) yields ˙ δx(t) = fˆx [t]δx(t) + fˆy [t]δy(t) + fu0 [t](c0u [t])+ k(t), 0ny = gˆx [t]δx(t) + gˆy [t]δy(t) + gu0 [t](c0u [t])+ k(t),
δx(t0 ) = 0nx ,
(4.25) (4.26)
where fˆx [t] := fx0 [t] − fu0 [t](c0u [t])+ c0x [t], fˆy [t] := f 0 [t] − f 0 [t](c0 [t])+ c0 [t], gˆx [t] := gˆy [t] :=
y u u y 0 0 0 + 0 gx [t] − gu [t](cu [t]) cx [t], gy0 [t] − gu0 [t](c0u [t])+ c0y [t].
Notice, that gˆy [t] is assumed to be non-singular. Using the solution formula in Lemma 3.2, the solution of (4.25),(4.26) can be written as Z t ˆ δx(t) = Φ(t) w(τ )> k(τ )dτ (4.27) t0
δy(t) = −ν1 (t)> δx(t) − ν2 (t)> k(t)
(4.28)
where −1 ˆ˙ ˆ Φ(t) = fˆx [t] − fˆy [t] (ˆ gy [t]) gˆx [t] Φ(t),
ˆ 0 ) = In Φ(t x
and −1 ˆ −1 (t) fu0 [t] − fˆy [t] (ˆ w(t)> := Φ gy [t]) gu0 [t] (c0u [t])+ , −1
gˆx [t],
−1
gu0 [t](c0u [t])+ .
ν1 (t)> := (ˆ gy [t]) >
ν2 (t) := (ˆ gy [t])
Equations (4.24), (4.27), and (4.28) and integration by parts yields Z tf −η1∗ (k(·)) = Hu0 [t](c0u [t])+ I + c0y [t]ν2 (t)> k(t)dt t0
Z
tf
−
ˆ Hu0 [t](c0u [t])+ cˆ[t]Φ(t)
t0
Z
tf
=
Z
t
>
w(τ ) k(τ )dτ
dt
t0
Hu0 [t](c0u [t])+ I + c0y [t]ν2 (t)> k(t)dt
t0
Z
tf
Z
− t0
tf
ˆ )dτ Hu0 [τ ](c0u [τ ])+ cˆ[τ ]Φ(τ
w(t)> k(t)dt
t
where −1
cˆ[t] := c0x [t] − c0y [t] (ˆ gy [t])
gˆx [t] = c0x [t] − c0y [t]ν1 (t)> .
With the definition Z tf > 0 0 + ˆ η(t) := Hu [τ ](cu [τ ]) cˆ[τ ]Φ(τ )dτ w(t)> − Hu0 [t](c0u [t])+ I + c0y [t]ν2 (t)> t
16
M. GERDTS
we thus obtained the representation η1∗ (k(·))
tf
Z
η(t)> k(t)dt,
= t0
where η is an element of L∞ ([t0 , tf ], Rnc ). Introducing this representation into (4.9), setting h2 (·) = Θ, using integration by parts and the solution formulas (4.3), (4.4) leads to the following representation of the functional λ∗1 : Z t Z tf Z tf Φ(τ )−1 h1 (τ )dτ dt −λ∗1 (h1 (·)) = p1 (t)> h1 (t)dt + η(t)> ν3 (t)> Φ(t) t0 tf
Z
t0 tf
Z
>
=
t0 tf
Z
p1 (t) h1 (t)dt + t0 Z tf
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ t0
Φ(t)−1 h1 (t)dt
t
λ1 (t)> h1 (t)dt,
= t0
where >
>
Z
tf
λ1 (t) := p1 (t) +
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ
Φ(t)−1
t
and −1 0 ν3 (t)> := c0x [t] − c0y [t] gy0 [t] gx [t]. Thus, we obtained the representation Z λ∗1 (h1 (·)) = −
tf
λ1 (t)> h1 (t)dt,
t0
where λ1 is an element of BV ([t0 , tf ], Rnx ). Introducing the representation of η1∗ into (4.9), setting h1 (·) = Θ, using integration by parts and the solution formulas (4.3), (4.4) leads to the following representation of the functional λ∗2 : Z tf −1 ∗ −λ2 (h2 (·)) = − −p2 (t)> + η(t)> c0y [t] gy0 [t] h2 (t)dt t0 tf
Z −
t0 Z tf
=− t0 tf
Z
η(t)> ν3 (t)> Φ(t)
−1 Φ(τ )−1 fy0 [τ ] gy0 [τ ] h2 (τ )dτ
dt
−1 −p2 (t)> + η(t)> c0y [t] gy0 [t] h2 (t)dt
Z
tf
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ t0 tf
t0
− Z
t
Z
−1 Φ(t)−1 fy0 [t] gy0 [t] h2 (t)dt,
t
λ2 (t)> h2 (t)dt,
= t0
where >
>
λ2 (t) := p2 (t) −
η(t)> c0y [t]
−1 gy0 [t]
Z −
tf
>
>
η(τ ) ν3 (τ ) Φ(τ )dτ t
−1 Φ(t)−1 fy0 [t] gy0 [t] .
LOCAL MINIMUM PRINCIPLE
17
Thus, we obtained the representation λ∗2 (h2 (·)) = −
Z
tf
λ2 (t)> h2 (t)dt,
t0
where λ2 is an element of L∞ ([t0 , tf ], Rny ). Remark 4.3. The regularity assumptions (4.14) and (4.15) can be replaced by the assumption that the matrix 0 gy [t] gu0 [t] c0y [t] c0u [t] is non-singular a.e. in [t0 , tf ]. Then, Equations (4.20) and (4.22) can be solved w.r.t. δy and δu and the proof can be adapted. The remarkable result in the preceeding corollaries is, that they provide useful representations of the functionals λ∗1 , λ∗2 , and η1∗ . Originally, these functionals are elements of the dual space of L∞ , which has a very complicated structure. The exploitation of the fact, that λ∗1 , λ∗2 , and η1∗ are multipliers in an optimization problem, showed that these functionals actually are more regular. 5. Local Minimum Principles. Theorem 3.6 and Corollary 4.1 yield the following necessary conditions for the optimal control problem 1.1. The Hamilton function is defined by > H(t, x, y, u, λ1 , λ2 , l0 ) := l0 f0 (t, x, y, u) + λ> 1 f (t, x, y, u) + λ2 g(t, x, y, u).
(5.1)
Theorem 5.1 (Local Minimum Principle). Let the following assumptions be fulfilled for the optimal control problem 1.1. (i) Let the functions ϕ, f0 , f, g, s, ψ be continuous w.r.t. all arguments and continuously differentiable w.r.t. x, y, and u. (ii) Let U ⊆ Rnu be a closed and convex set with non-empty interior. (iii) Let (ˆ x, yˆ, u ˆ) ∈ X be a weak local minimum of the optimal control problem. (iv) Let Assumption 3.3 be valid. (v) Let there be no mixed control-state constraints (1.5) in the optimal control problem 1.1. Then there exist multipliers l0 ∈ R, σ ∈ Rnψ , λ1 ∈ BV ([t0 , tf ], Rnx ), λ2 ∈ L∞ ([t0 , tf ], Rny ), and µ ∈ N BV ([t0 , tf ], Rns ) such that the following conditions are satisfied: (i) l0 ≥ 0, (l0 , σ, λ1 , λ2 , µ) 6= Θ, (ii) Adjoint equations: Z
tf
Hx0 (τ, x ˆ(τ ), yˆ(τ ), u ˆ(τ ), λ1 (τ ), λ2 (τ ), l0 )> dτ t ns Z tf X + s0i,x (τ, x ˆ(τ ))> dµi (τ ) in [t0 , tf ], t i=1 Hy0 (t, x ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), l0 )> a.e. in [t0 , tf ].
λ1 (t) = λ1 (tf ) +
0ny =
(5.2) (5.3)
(iii) Transversality conditions: λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 , >
λ1 (tf ) =
l0 ϕ0xf
+σ
>
ψx0 f .
(5.4) (5.5)
18
M. GERDTS
(iv) Optimality conditions: Almost everywhere in [t0 , tf ] for all u ∈ U it holds Hu0 (t, x ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), l0 )(u − u ˆ(t)) ≥ 0.
(5.6)
(v) Complementarity condition: µi is monotonically increasing on [t0 , tf ] and constant on every interval (t1 , t2 ) with t1 < t2 and si (t, x ˆ(t)) < 0 for all t ∈ (t1 , t2 ). Proof. Under the above assumptions, Corollary 4.1 (with p1 = λ1 , p2 = λ2 ) guarantees the existence of λ1 ∈ BV ([t0 , tf ], Rnx ) and λ2 ∈ L∞ ([t0 , tf ], Rny ) such that Z tf ∗ λ1 (h1 (·)) = − λ1 (t)> h1 (t)dt, (5.7) λ∗2 (h2 (·)) = −
t0 tf
Z
λ2 (t)> h2 (t)dt,
(5.8)
t0
holds for every h1 ∈ L∞ ([t0 , tf ], Rnx ) and h2 ∈ L∞ ([t0 , tf ], Rny ). (a) Equation (3.15) is equivalent with
Z
tf
+
l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) Z tf ns Z tf X ˙ Hx0 [t]δx(t)dt + s0i,x [t]δx(t)dµi (t) − λ1 (t)> δx(t)dt =
t0
t0
i=1
0
t0
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). Application of the computation rules for Stieltjes integrals yields l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) Z tf Z tf ns Z tf X 0 0 + Hx [t]δx(t)dt + si,x [t]δx(t)dµi (t) − λ1 (t)> dδx(t) = 0, t0
t0
i=1
t0
where Z
tf
>
λ1 (t) dδx(t) := t0
nx Z X i=1
tf
λ1,i (t)dδxi (t).
t0
Integration by parts of the last term leads to l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf ) Z tf Z tf ns Z tf X + Hx0 [t]δx(t)dt + s0i,x [t]δx(t)dµi (t) + δx(t)> dλ1 (t)
=
0
l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf ) ! Z tf Z tf ns Z tf X > 0 > 0 > + δx(t) d − Hx [τ ] dτ − si,x [τ ] dµi (τ ) + λ1 (t) =
0
t0
i=1
t0
t0
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This is equivalent with
t0
t
i=1
t
19
LOCAL MINIMUM PRINCIPLE
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This implies λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 , >
λ1 (tf ) =
l0 ϕ0xf
+σ tf
Z C = λ1 (t) −
>
(5.9)
ψx0 f ,
(5.10)
Hx0 [τ ]> dτ −
t
ns Z tf X i=1
s0i,x [τ ]> dµi (τ )
(5.11)
t
for some constant vector C. Evaluation of the last equation at t = tf yields C = λ1 (tf ), which yields (5.2), (5.4), and (5.5). (b) Equation (3.12) is equivalent with Z
tf
Hy0 [t]δy(t)dt = 0
t0
for all δy ∈ L∞ ([t0 , tf ], Rny ). This implies (5.3). (c) Introducing (5.7),(5.8) into (3.13) leads to the variational inequality Z
tf
Hu0 [t](u(t) − u ˆ(t))dt ≥ 0
t0
for all u ∈ Uad . This implies Hu0 [t](u − u ˆ(t)) ≥ 0 for all u ∈ U for almost all t ∈ [t0 , tf ], cf. Kirsch et al. [13], p. 102. This is the optimality condition. (d) According to Theorem 3.6, (3.8), (3.9) it holds η2∗ ∈ K2+ , i.e. η2∗ (z) =
ns Z X i=1
tf
zi (t)dµi (t) ≥ 0
t0
for all z ∈ K2 = {z ∈ C([t0 , tf ], Rns ) | z(t) ≥ 0ns in [t0 , tf ]}. This implies, that µi is monotonically increasing. Finally, the condition η2∗ (s(·, x ˆ(·))) = 0, i.e. η2∗ (s(·, x ˆ(·)))
=
ns Z X i=1
tf
si (t, x ˆ(t))dµi (t) = 0,
t0
together with the monotonicity of µi implies that µi is constant in intervals with si (t, x ˆ(t)) < 0. Likewise, Theorem 3.6 and Corollary 4.2 yield the following necessary conditions for the optimal control problem 1.1. The augmented Hamilton function is defined by ˆ x, y, u, λ1 , λ2 , η, l0 ) := H(t, x, y, u, λ1 , λ2 , l0 ) + η > c(t, x, y, u) H(t,
(5.12)
Theorem 5.2 (Local Minimum Principle). Let the following assumptions be fulfilled for the optimal control problem 1.1. (i) Let the functions ϕ, f0 , f, g, c, s, ψ be continuous w.r.t. all arguments and continuously differentiable w.r.t. x, y, and u.
20
M. GERDTS
(ii) Let (ˆ x, yˆ, u ˆ) ∈ X be a weak local minimum of the optimal control problem. (iii) Let rank (c0u (t, x ˆ(t), yˆ(t), u ˆ(t))) = nc almost everywhere in [t0 , tf ], cf. Corollary 4.2, (4.14). (iv) Let the matrix gy0 [t] − gu0 [t](c0u [t])+ c0y [t]
(5.13)
be non-singular almost everywhere in [t0 , tf ], where (c0u [t])+ := c0u [t]> c0u [t]c0u [t]>
−1
.
(5.14)
denotes the pseudo-inverse of c0u [t]. (v) Let Assumption 3.3 be valid. (vi) Let U = Rnu . Then there exist multipliers l0 ∈ R, σ ∈ Rnψ , λ1 ∈ BV ([t0 , tf ], Rnx ), λ2 ∈ L∞ ([t0 , tf ], Rny ), η ∈ L∞ ([t0 , tf ], Rnc ), and µ ∈ N BV ([t0 , tf ], Rns ) such that the following conditions are satisfied: (i) l0 ≥ 0, (l0 , σ, λ1 , λ2 , η, µ) 6= Θ, (ii) Adjoint equations: Z
tf
ˆ 0 (τ, x H ˆ(τ ), yˆ(τ ), u ˆ(τ ), λ1 (τ ), λ2 (τ ), η(τ ), l0 )> dτ x t ns Z tf X s0i,x (τ, x ˆ(τ ))> dµi (τ ) in [t0 , tf ], (5.15) + t i=1 ˆ 0 (t, x H ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), η(t), l0 )> a.e. in [t0 , tf ]. (5.16) y
λ1 (t) = λ1 (tf ) +
0ny =
(iii) Transversality conditions: λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 , >
λ1 (tf ) =
l0 ϕ0xf
+σ
>
ψx0 f .
(5.17) (5.18)
(iv) Optimality conditions: It holds ˆ u0 (t, x H ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), η(t), l0 ) = 0nu
(5.19)
a.e. in [t0 , tf ]. (v) Complementarity conditions: It holds η(t)> c(t, x ˆ(t), yˆ(t), u ˆ(t)) = 0,
η(t) ≥ 0nc .
almost everywhere in [t0 , tf ]. µi is monotonically increasing on [t0 , tf ] and constant on every interval (t1 , t2 ) with t1 < t2 and si (t, x ˆ(t)) < 0 for all t ∈ (t1 , t2 ). Proof. Corollary 4.2 yields the existence of the functions λ1 , λ2 , η and provides representations of the functionals λ∗1 , λ∗2 , and η1∗ .
21
LOCAL MINIMUM PRINCIPLE
(a) Equation (3.13) becomes tf
Z
ˆ u0 [t]δu(t)dt = 0 H
t0
for all δu ∈ L∞ ([t0 , tf ], Rnu ). This implies (5.19). (b) Equation (3.15) leads to Z tf ˆ x0 [t]δx(t)dt l0 ϕ0x0 + σ > ψx0 0 δx(t0 ) + l0 ϕ0xf + σ > ψx0 f δx(tf ) + H t0 Z tf ns Z tf X 0 ˙ + si,x [t]δx(t)dµi (t) − λ1 (t)> δx(t)dt = t0
i=1
0
t0
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). Application of the computation rules for Stieltjes integrals yields Z tf 0 > 0 0 > 0 ˆ x0 [t]δx(t)dt l0 ϕx0 + σ ψx0 δx(t0 ) + l0 ϕxf + σ ψxf δx(tf ) + H t0 Z tf ns Z tf X s0i,x [t]δx(t)dµi (t) − + λ1 (t)> dδx(t) = 0. t0
i=1
t0
Integration by parts of the last term leads to l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf ) Z tf Z tf ns Z tf X 0 0 ˆ + Hx [t]δx(t)dt + si,x [t]δx(t)dµi (t) + δx(t)> dλ1 (t)
=
0,
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This is equivalent with l0 ϕ0x0 + σ > ψx0 0 + λ1 (t0 )> δx(t0 ) + l0 ϕ0xf + σ > ψx0 f − λ1 (tf )> δx(tf ) ! Z tf Z tf ns Z tf X > 0 > 0 > ˆ + δx(t) d − Hx [τ ] dτ − si,x [τ ] dµi (τ ) + λ1 (t) =
0,
t0
i=1
t0
t
t0
t0
i=1
t
for all δx ∈ W 1,∞ ([t0 , tf ], Rnx ). This implies λ1 (t0 )> = − l0 ϕ0x0 + σ > ψx0 0 , λ1 (tf )> =
l0 ϕ0xf + σ > ψx0 f , Z tf ns Z X 0 > ˆ C = λ1 (t) − Hx [τ ] dτ − t
i=1
tf
s0i,x [τ ]> dµi (τ )
t
for some constant vector C. Evaluation of the last equation at t = tf yields C = λ1 (tf ), which proves (5.15). (c) Equation (3.12) becomes Z tf ˆ y0 [t]δy(t)dt = 0 H t0
for all δy ∈ L∞ ([t0 , tf ], Rny ). This implies (5.16).
22
M. GERDTS
(d) From η1∗ ∈ K1+
Z
tf
η(t)> z(t)dt ≥ 0 ∀z ∈ K1
⇔ t0
and Z
η1∗ (c(·, x ˆ(·), yˆ(·), u ˆ(·))) = 0
tf
⇔
η(t)> c[t]dt = 0
t0
we conclude that η(t) ≥ 0nc ,
η(t)> c[t] = 0,
a.e. in [t0 , tf ].
According to Theorem 3.6, (3.8), (3.9) it holds η2∗ ∈ K2+ , i.e. η2∗ (z)
=
ns Z X i=1
tf
zi (t)dµi (t) ≥ 0
t0
for all z ∈ K2 = {z ∈ C([t0 , tf ], Rns ) | z(t) ≥ 0ns in [t0 , tf ]}. This implies, that µi is monotonically increasing. Finally, the condition η2∗ (s(·, x ˆ(·))) = 0, i.e. η2∗ (s(·, x ˆ(·))) =
ns Z X i=1
tf
si (t, x ˆ(t))dµi (t) = 0,
t0
together with the monotonicity of µi implies that µi is constant in intervals with si (t, x ˆ(t)) < 0. The following considerations apply to both, Theorem 5.1 and Theorem 5.2 and ˆ respectively. Hence, we restrict the differ only in the Hamilton functions H and H, discussion to the situation of Theorem 5.1. The multiplier µ is of bounded variation. Hence, it has at most countably many jump points and µ can be expressed as µ = µa + µd + µs , where µa is absolutely continuous, µd is a jump function, and µs is singular (continuous, non-constant, µ˙ s = 0 a.e.). Hence, the adjoint equation (5.2) can be written as Z tf λ1 (t) = λ1 (tf ) + Hx0 (τ, x ˆ(τ ), yˆ(τ ), u ˆ(τ ), λ1 (τ ), λ2 (τ ), l0 )> dτ t Z tf ns Z tf X s0i,x (τ, x ˆ(τ ))> dµi,a (τ ) + s0i,x (τ, x ˆ(τ ))> dµi,d (τ ) + t t i=1 Z tf 0 > + si,x (τ, x ˆ(τ )) dµi,s (τ ) t
for all t ∈ [t0 , tf ]. Notice, that λ1 is continuous from the right in (t0 , tf ) since µ is normalized and thus continuous from the right in (t0 , tf ).
23
LOCAL MINIMUM PRINCIPLE
Let {tj }, j ∈ J be the jump points of µ. Then, at every jump point tj ∈ (t0 , tf ) it holds ! Z tf Z tf 0 > 0 > lim si,x (τ, x ˆ(τ )) dµi,d (τ ) − si,x (τ, x ˆ(τ )) dµi,d (τ ) ε↓0
tj −ε
tj
= −s0i,x (τ, x ˆ(τ ))> (µi,d (tj ) − µi,d (tj −)) . Since µa is absolutely continuous and µs is continuous we obtain the jump-condition λ1 (tj ) − λ1 (tj −) = −
ns X
s0i,x (tj , x ˆ(tj ))> (µi (tj ) − µi (tj −)) ,
j ∈ J , tj ∈ (t0 , tf ).
i=1
In order to derive a differential equation for λ1 we need the subsequent auxiliary result. Corollary 5.3. Let w : [t0 , tf ] → R be continuous and µ monotonically increasing on [t0 , tf ]. Let µ be differentiable at t ∈ [t0 , tf ]. Then it holds d dt
Z
t
w(s)dµ(s) = w(t)µ(t). ˙ t0
Proof. Define Z
t
W (t) =
w(s)dµ(s). t0
The mean-value theorem for Riemann-Stieltjes integrals yields Z t+h W (t + h) − W (t) = w(s)dµ(s) = w(ξh )(µ(t + h) − µ(t)) t
for some ξh ∈ [t, t + h]. Hence, it follows W (t + h) − W (t) w(ξh ) · (µ(t + h) − µ(t)) = lim = w(t)µ(t), ˙ h→0 h→0 h h lim
where ξh lies between t and t + h and thus ξh → t for h → 0. Furthermore, since every function of bounded variation is differentiable almost everywhere, µ and λ1 are differentiable almost everywhere. Putting everything together, we proved Corollary 5.4. Let the assumptions of Theorem 5.1 be fulfilled. Then, λ1 is differentiable almost everywhere in [t0 , tf ] with λ˙ 1 (t) = −Hx0 (t, x ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), l0 )> −
ns X
s0i,x (t, x ˆ(t))> µ˙ i (t).
(5.20)
i=1
Furthermore, the jump conditions λ1 (tj ) − λ1 (tj −) = −
ns X
s0i,x (tj , x ˆ(tj ))> (µi (tj ) − µi (tj −))
i=1
hold at every point tj ∈ (t0 , tf ) of discontinuity of the multiplier µ.
(5.21)
24
M. GERDTS
Notice, that the component µ˙ i in (5.20) can be replaced by the absolutely continuous component µ˙ i,a , since the derivatives µd and µs vanish almost everywhere. Similarly, µi in (5.21) can be replaced by the discrete part µd,i , since µa and µs are continuous. Notice furthermore, that (5.20) and (5.21) provide the absolutely continuous respectively discrete components of λ1 , whereas the singular component of λ1 occurs in (5.2) only. A special case arises, if no state constraints are present. Then, the adjoint variable λ1 is even absolutely continuous, i.e. λ1 ∈ W 1,∞ ([t0 , tf ], Rnx ), and the adjoint equations (5.2)-(5.3) become λ˙ 1 (t) = −Hx0 (t, x ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), l0 )> a.e. in [t0 , tf ], 0ny = Hy0 (t, x ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), l0 )> a.e. in [t0 , tf ].
(5.22) (5.23)
The adjoint equations (5.22) and (5.23) form a DAE system of index one for λ1 and λ2 , where λ1 is the differential variable and λ2 denotes the algebraic variable. This follows from (5.23), which is given by > > > 0 0ny = l0 f0,y [t] + fy0 [t] λ1 (t) + gy0 [t] λ2 (t). Since gy0 [t] is non-singular, we obtain −> > > 0 λ2 (t) = − gy0 [t] l0 f0,y [t] + fy0 [t] λ1 (t) . In this section we concentrated on local minimum principles only. The term ‘local’ is due to the fact, that the optimality conditions (5.6) and (5.19), respectively, can be interpreted as necessary conditions for a local minimum of the Hamilton function and the augmented Hamilton function, respectively. However, there are also global minimum principles. In a global minimum principle the optimality condition is given by ˆ x ˆ x H(t, ˆ(t), yˆ(t), u ˆ(t), λ1 (t), λ2 (t), η(t), l0 ) ≤ H(t, ˆ(t), yˆ(t), u, λ1 (t), λ2 (t), η(t), l0 ) for all u ∈ U with c(t, x ˆ(t), yˆ(t), u) ≤ 0 almost everywhere in [t0 , tf ]. If the Hamilton function and the function c is convex w.r.t. the control u, then both conditions are equivalent. Furthermore, the main important difference between local and global minimum principle is, that U can be an arbitrary subset of Rnu in the global case, e.g. a discrete set. In our approach we had to assume that U is a convex set with nonempty interior. Proofs for global minimum resp. maximum principles in the context of ordinary differential equations subject to pure state constraints can be found in, e.g., Girsanov [6] and Ioffe and Tihomirov [11], pp. 147-159, 241-253. We favored the statement of necessary conditions in terms of local minimum principles because these conditions are closer to the finite dimensional case. A connection arises, if the infinite dimensional optimal control problem is discretized and transformed into a finite dimensional optimization problem. The formulation of the necessary Fritz-John conditions leads to the discrete minimum principle. These conditions are comparable to the infinite dimensional conditions. This is different for the global minimum principle. In the finite dimensional case a comparable minimum principle only holds approximately, cf. Mordukhovich [25], or under additional convexity like assumptions, cf. Ioffe and Tihomirov [11], p. 278.
25
LOCAL MINIMUM PRINCIPLE
6. Regularity. We state conditions which ensure that the multiplier l0 is not zero and without loss of generality can be normalized to one. Again, we consider the optimal control problem 1.1 and the optimization problem 3.1, respectively. The functions ϕ, f0 , f, g, ψ, c, s are assumed to be continuous w.r.t. all arguments and continuously differentiable w.r.t. x, y, and u. As mentioned before, this implies that F is Fr´echet-differentiable and G and H are continuously Fr´echet-differentiable. According to Corollary 2.2 the following Mangasarian-Fromowitz conditions imply l0 = 1: (i) H 0 (ˆ x, yˆ, u ˆ) is surjective. (ii) There exists some (δx, δy, δu) ∈ int(S − {(ˆ x, yˆ, u ˆ)}) with H 0 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = ΘZ , G(ˆ x, yˆ, u ˆ) + G0 (ˆ x, yˆ, u ˆ)(δx, δy, δu) ∈ int(K). Herein, the interior of S = W 1,∞ ([t0 , tf ], Rnx ) × L∞ ([t0 , tf ], Rny ) × Uad is given by int(S) = {(x, y, u) ∈ X | ∃ε > 0 : Bε (u(t)) ⊆ U a.e. in [t0 , tf ]} , where Bε (u) denotes the open ball with radius ε around u. The interior of K1 is given by int(K1 ) = {z ∈ L∞ ([t0 , tf ], Rnc ) | ∃ε > 0 : Uε (z) ⊆ K1 } = {z ∈ L∞ ([t0 , tf ], Rnc ) | ∃ε > 0 : zi (t) ≥ ε a.e. in [t0 , tf ], i = 1, . . . , nc } . The interior of K2 is given by int(K2 ) = {z ∈ C([t0 , tf ], Rns ) | z(t) > 0ns in [t0 , tf ]} . Notice, that the assumption int(K) 6= ∅ in Corollary 2.2 is satisfied for Problem 3.1. A sufficient condition for (i) to hold is given by Lemma 6.1. Let Assumption 3.3 be valid and let rank ψx0 0 Φ(t0 ) + ψx0 f Φ(tf ) = nψ , (6.1) where Φ is the fundamental solution of the homogeneous linear differential equation ˙ Φ(t) = A(t)Φ(t),
Φ(t0 ) = Inx ,
t ∈ [t0 , tf ]
and −1 0 A(t) := fx0 [t] − fy0 [t] gy0 [t] gx [t]. Then H 0 (ˆ x, yˆ, u ˆ) in Problem 3.1 is surjective. Proof. Appendix A Remark 6.2. The rank condition (6.1) together with Lemma 3.2 implies that the DAE (1.1)-(1.2) is completely output controllable. Condition (ii) is satisfied if there exist δx ∈ W 1,∞ ([t0 , tf ], Rnx ), δy ∈ L∞ ([t0 , tf ], Rny ), δu ∈ int(Uad − {ˆ u}), and ε > 0 satisfying c[t] + c0x [t]δx(t) + c0y [t]δy(t) + c0u [t]δu(t) ≤ −ε · e, a.e. in [t0 , tf ], s[t] + s0x [t]δx(t) < 0ns , in [t0 , tf ], ˙ f 0 [t]δx(t) + f 0 [t]δy(t) + f 0 [t]δu(t) − δx(t) = 0n , a.e. in [t0 , tf ], x
y
u
x
gx0 [t]δx(t) + gy0 [t]δy(t) + gu0 [t]δu(t) = 0ny , ψx0 0 δx(t0 ) + ψx0 f δx(tf ) = 0nψ ,
a.e. in [t0 , tf ],
(6.2) (6.3) (6.4) (6.5) (6.6)
26
M. GERDTS
where e = (1, . . . , 1)> ∈ Rnc . Hence, we conclude Theorem 6.3. Let the assumptions of Theorems 3.6, 5.1 or 5.2, and Lemma 6.1 be fulfilled. Furthermore, let there exist δx ∈ W 1,∞ ([t0 , tf ], Rnx ), δy ∈ L∞ ([t0 , tf ], Rny ), and δu ∈ int(Uad − {ˆ u}) satisfying (6.2)-(6.6). Then it holds l0 = 1 in Theorems 3.6 and 5.1 or 5.2, respectively. 7. Conclusions and outlook. The presented local minimum principles for optimal control problems subject to index-1 differential-algebraic equations are not only of theoretical interest but gives rise to numerical solution methods for such problems. The so-called indirect approach intends to fulfill the necessary conditions for the optimal control problem numerically and thus produces candidates for an optimal solution. Notice, that the evaluation of the local minimum principle will lead to a multi-point boundary value problem, at least under suitable simplifying assumptions such as, e.g., that the structure of active and inactive state constraints is known and that the singular part of the multiplier µ vanishes, cf. Corollary 5.4. Even for so-called direct methods, which are based on a discretization of the optimal control problem, cf. Gerdts [4], the local minimum principle is of great importance in view of the post-optimal approximation of adjoints. In the future, it is important to extend these necessary conditions also to higher index differential-algebraic equations, which do not allow to solve the algebraic equation for y directly. First attempts in this direction show, that it is possible to obtain likewise results for special classes of index-2 differential-algebraic equations. Appendix A. Proofs of auxiliary lemmata. Proof of Lemma 3.2: Proof. We exploit the non-singularity of B2 (t) and the essential boundedness of B2 (t)−1 . Equation (3.2) can be solved w.r.t. y: y(t) = −B2 (t)−1 (A2 (t)x(t) + h2 (t)) . Introducing this expression into (3.1) yields x(t) ˙ = A1 (t) − B1 (t)B2 (t)−1 A2 (t) x(t) + h1 (t) − B1 (t)B2 (t)−1 h2 (t) = A(t)x(t) + h(t). For given x(t0 ) = x0 we are in the situation of Hermes and Lasalle [8], p. 36, and the assertion follows likewise. Part (b) exploits the solution formulas in (a). The boundary conditions (3.6) are satisfied, if b = C0 x(t0 ) + Cf x(tf ) Z = C0 x(t0 ) + Cf Φ(tf ) x(t0 ) +
tf
Φ−1 (τ )h(τ )dτ
t0
Z
tf
Φ−1 (τ )h(τ )dτ.
= (C0 + Cf Φ(tf )) x(t0 ) + Cf Φ(tf ) t0
Rearranging terms and exploiting Φ(t0 ) = Inx yields Z
tf
(C0 Φ(t0 ) + Cf Φ(tf )) x(t0 ) = b − Cf Φ(tf ) t0
Φ−1 (τ )h(τ )dτ.
LOCAL MINIMUM PRINCIPLE
27
This equation is solvable for every b ∈ Rr , if the matrix C0 Φ(t0 ) + Cf Φ(tf ) is of rank r. Then, for every b ∈ Rr there exists a x(t0 ) satisfying (3.6). Application of part (a) completes the proof. Proof of Lemma 3.5 (cf. Ljusternik and Sobolev [18], Lempio [17]): Proof. Suppose that im(T ) is not closed. Then there exists a sequence (yi , zi ) ∈ im(T ) with limi→∞ (yi , zi ) = (y0 , z0 ) 6∈ im(T ). Then, there exists a (algebraic) hyperplane, which contains im(T ) but not (y0 , z0 ). Hence, there exist a linear functional ly ∈ Y 0 and a continuous linear functional lz ∈ (Rn )∗ , not both zero, with (x ∈ X)
ly (T1 (x)) + lz (T2 (x)) = 0, ly (y0 ) + lz (z0 ) 6= 0.
Notice, that ly is not necessarily continuous. In fact, we will show that ly is actually continuous. Therefore, let ηi ∈ Y be an arbitrary sequence converging to zero in Y . Unfortunately, there may be many points in the pre-image of ηi under the mapping T1 . To circumvent this problem, instead we consider the factor space X/ker(T1 ) whose elements [x] are defined by [x] := x+ker(T1 ) for x ∈ X. The space X/ker(T1 ) endowed with the norm k[x]kX/ker(T1 ) := inf{ky − xkX | y ∈ ker(T1 )} is a Banach space. Notice that ker(T1 ) is a closed subspace of X. If we consider T1 as a mapping from the factor space X/ker(T1 ) onto Y , then the inverse operator of this mapping is continuous, because T1 is surjective and X and Y are Banach spaces. Hence, to each ηi ∈ Y there corresponds an element Wi of X/ker(T1 ). Since the inverse operator is continuous and ηi is a null-sequence, the sequence Wi converges to zero. Furthermore, we can choose a representative ξi ∈ Wi such that kξi kX ≤ 2kWi kX/ker(T1 ) ,
T1 (ξi ) = ηi
hold for every i ∈ N. Since Wi is a null-sequence the same holds for ξi by the first inequality. This yields lim ly (ηi ) = lim ly (T1 (ξi )) = lim −lz (T2 (ξi )) = 0,
i→∞
i→∞
i→∞
since ξi → ΘX and T2 and lz are continuous. This shows, that ly is actually continuous in ΘX and hence on X. In particular, we obtain 0 = lim (ly (yi ) + lz (zi )) = ly i→∞
lim yi + lz lim zi = ly (y0 ) + lz (z0 )
i→∞
i→∞
in contradiction to 0 6= ly (y0 ) + lz (z0 ). Proof of Lemma 6.1: Proof. Let h1 ∈ L∞ ([t0 , tf ], Rnx ), h2 ∈ L∞ ([t0 , tf ], Rny ), and h3 ∈ Rnψ be given. Consider the boundary value problem H10 (ˆ x, yˆ, u ˆ)(δx, δy, δu)(t) = h1 (t), H20 (ˆ x, yˆ, u ˆ)(δx, δy, δu)(t) = h2 (t), 0 H3 (ˆ x, yˆ, u ˆ)(δx, δy, δu) = h3 .
a.e. in [t0 , tf ], a.e. in [t0 , tf ],
28
M. GERDTS
In its components the boundary value problem becomes ˙ −δx(t) + fx0 [t]δx(t) + fy0 [t]δy(t) + fu0 [t]δu(t) = h1 (t), gx0 [t]δx(t) + gy0 [t]δy(t) + gu0 [t]δu(t) = h2 (t), 0 ψx0 δx(t0 ) + ψy0 0 δy(t0 ) + ψx0 f δx(tf ) + ψy0 f δy(tf ) = −h3 .
a.e. in [t0 , tf ], a.e. in [t0 , tf ],
By the rank assumption and Assumption 3.3 all assumptions of Lemma 3.2 are satisfied and the boundary value problem is solvable. This shows the surjectivity of the mapping H 0 (ˆ x, yˆ, u ˆ). REFERENCES [1] A. E. Bryson and Y.-C. Ho, Applied Optimal Control, Hemisphere Publishing Corporation, Washington, 1975. [2] M. de Pinho and R. B. Vinter, Necessary Conditions for Optimal Control Problems Involving Nonlinear Differential Algebraic Equations, Journal of Mathematical Analysis and Applications, 212 (1997), pp. 493–516. [3] E. N. Devdariani and Y. S. Ledyaev, Maximum Principle for Implicit Control Systems, Applied Mathematics and Optimization, 40 (1999), pp. 79–103. [4] M. Gerdts, Direct Shooting Method for the Numerical Solution of Higher Index DAE Optimal Control Problems, Journal of Optimization Theory and Applications, 117 (2003), pp. 267– 294. , A moving horizon technique in vehicle simulation, ZAMM, 83 (2003), pp. 147–162. [5] [6] I. V. Girsanov, Lectures on Mathematical Theory of Extremum Problems, vol. 67 of Lecture Notes in Economics and Mathematical Systems, Berlin-Heidelberg-New York, 1972, Springer. [7] R. F. Hartl, S. P. Sethi, and G. Vickson, A Survey of the Maximum Principles for Optimal Control Problems with State Constraints, SIAM Review, 37 (1995), pp. 181–218. [8] H. Hermes and J. P. Lasalle, Functional Analysis and Time Optimal Control, vol. 56 of Mathematics in Science and Engineering, Academic Press, New York, 1969. [9] M. R. Hestenes, Calculus of variations and optimal control theory, John Wiley & Sons, New York, 1966. [10] H. Hinsberger, Ein direktes Mehrzielverfahren zur L¨ osung von Optimalsteuerungsproblemen mit großen, differential-algebraischen Gleichungssystemen und Anwendungen aus der Verfahrenstechnik, PhD thesis, Institut f¨ ur Mathematik, Technische Universit¨ at Clausthal, 1997. [11] A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, vol. 6 of Studies in Mathematics and its Applications, Amsterdam, New York, Oxford, 1979, North-Holland Publishing Company. [12] D. H. Jacobson, M. M. Lele, and J. L. Speyer, New Necessary Conditions of Optimality for Constrained Problems with State-Variable Inequality Constraints, Journal of Mathematical Analysis and Applications, 35 (1971), pp. 255–284. [13] A. Kirsch, W. Warth, and J. Werner, Notwendige Optimalit¨ atsbedingungen und ihre Anwendung, vol. 152 of Lecture Notes in Economics and Mathematical Systems, BerlinHeidelberg-New York, 1978, Springer. [14] H. W. Knobloch, Das Pontryaginsche Maximumprinzip f¨ ur Probleme mit Zustandsbeschr¨ ankungen i und ii, Zeitschrift f¨ ur Angewandte Mathematik und Mechanik, 55 (1975), pp. 545–556, 621–634. [15] E. Kreindler, Additional Necessary Conditions for Optimal Control with State-Variable Inequality Constraints, Journal of Optimization Theory and Applications, 38 (1982), pp. 241– 250. [16] S. Kurcyusz, On the Existence and Nonexistence of Lagrange Multipliers in Banach Spaces, Journal of Optimization Theory and Applications, 20 (1976), pp. 81–110. [17] F. Lempio, Tangentialmannigfaltigkeiten und infinite Optimierung, Habilitationsschrift, Universit¨ at Hamburg, Hamburg, 1972. [18] L. A. Ljusternik and W. I. Sobolew, Elemente der Funktionalanalysis, Verlag Harri Deutsch, Z¨ urich-Frankfurt/Main-Thun, 1976. [19] D. G. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, New YorkLondon-Sydney-Toronto, 1969.
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