An Exact Penalty Function Method for Continuous Inequality ...

J Optim Theory Appl (2011) 151:260–291 DOI 10.1007/s10957-011-9904-5

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem Bin Li · Chang Jun Yu · Kok Lay Teo · Guang Ren Duan

Published online: 1 September 2011 © Springer Science+Business Media, LLC 2011

Abstract In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with B. Li · G.R. Duan Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, China B. Li e-mail: [email protected] G.R. Duan e-mail: [email protected] B. Li · C.J. Yu · K.L. Teo () Department of Mathematics and Statistics, Curtin University, Perth, Australia e-mail: [email protected] C.J. Yu e-mail: [email protected] C.J. Yu Department of Mathematics, Shanghai University, Shanghai, China

J Optim Theory Appl (2011) 151:260–291

261

equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed. Keywords Optimal control · Terminal state equality constraints · Continuous state and control inequality constraints · Control parametrization · Time scaling transform · Exact penalty function · Numerical method 1 Introduction In many practical real-world optimal control problems, there are rigid requirements to be satisfied at every time point in the planning horizon. Such requirements are often expressed as equality terminal state constraints and continuous inequality constraints on the state and/or control. Thus, optimal control problems, with equality terminal state constraints and continuous inequality constraints on the state and/or control, have been studied intensively in the literature. See, for example [1–3]. In [1], necessary conditions for optimality have been derived for various types of constrained optimal control problems. However, many practical real-world problems are much too complex to admit analytical solutions by using these necessary conditions for optimality, and they can only be solved numerically. There are already some numerical methods, such as the discretization method [4–6], the non smooth Newton method [7–9] and the control parametrization method [2, 3], available in the literature. In particular, the control parametrization method has been used in conjunction with a time scaling transform [10] to solve a wide range of practical real-world problems. See, for example, [10] and [11]. In this method, the control function is approximated by a sequence of piecewise constant or piecewise linear functions. Then, the time scaling transform is used to convert the varying switching time points into fixed switching time points in a new time horizon. For the continuously inequality constraints, they are handled by the constraint transcription method [2], where the continuous inequality constraints are approximated by functions in integral form involving a smoothing parameter. These integral functions are either regarded as conventional inequality constraints or appended into the cost function by using the concept of the penalty function method to form a new cost function. For the first case, the original problem is approximated by a sequence of nonlinear optimization problems with inequality constraints in integral form, and each of which can be solved by standard constrained optimization methods, such as the sequential quadratic programming (SQP) approximation method. For the second case, we obtain a sequence of unconstrained nonlinear optimization problems and each of which is solvable by standard unconstrained nonlinear optimization techniques, such as conjugate gradient method or any quasiNewton method [11]. The constraint transcription method was originally developed in [2] to handle continuous inequality constraints on state variables only. The control parametrization technique and the time scaling transform are used to develop a computational method in [10] for optimal control problems subject to continuous inequality constraints on the state, where the continuous inequality constraints are handled by

262


this constraint transcription method. It is extended in [3] to the case where both the state and control are allowed to appear explicitly in the continuous inequality constraints. The nonlinear optimization problems, either constrained or unconstrained, are controlled by two parameters, where one controls the accuracy and the other controls the feasibility. However, the convergence for the method developed in [2, 3] and [10] may be slow. This happens because there are two parameters which need to be adjusted. More specifically, it is required to initiate a value for the accuracy parameter. Then, the feasibility parameter is required to be adjusted until the continuous inequality constraints are satisfied. The accuracy parameter is then reduced and followed by the adjustment of the feasibility parameter if the equality terminal state constraints and the continuous inequality constraints are not satisfied. The process is repeated until the required accuracy is achieved. Furthermore, there is no theoretical result showing that a local optimal solution of the nonlinear constrained optimization problem is a local optimal solution of the optimal parameter selection problem with equality terminal state constraints and continuous inequality constraints on the state and control. In this paper, we present a computational approach based on an exact penalty function method [12] for solving a class of optimal control problems subject to equality terminal state constraints and continuous inequality constraints on the state and control variables. After the control parametrization, together with a time scaling transformation, the problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous inequality constraints on the state and control. The new exact penalty functions, developed in [9] and [12], are constructed for these terminal equality constraints and continuous inequality constraints. They are appended to the cost function to form an augmented cost function, giving rise to an unconstrained optimal parameter selection problem. The convergence analysis shows that, for a sufficiently large penalty parameter, a local minimizer of the unconstrained optimization problem is a local minimizer of the optimal parameter selection problem with terminal equality constraints and continuous inequality constraints. The relationships between the approximate constrained optimal parameter selection problems and the original optimal control problem are also discussed. Finally, the method proposed is applied to solve three nontrivial optimal control problems. From the numerical results obtained, we see that the method proposed is effective.

2 Problem Statement Consider a dynamical system defined on [0, T ], dx(t) = f t, x(t), u(t) , dt

t ∈ (0, T ],

(1a)

with initial and terminal conditions x(0) = x 0 ,

(1b)

x(T ) = x f ,

(1c)


263

respectively, where T is the terminal time and x = [x1 , . . . , xn ] ∈ Rn

and u = [u1 , . . . , ur ] ∈ Rr

are, respectively, state and control vectors, while f = [f1 , . . . , fn ] ∈ Rn . We assume that the following conditions be satisfied. (A.1) f is continuously differentiable with respect to all it arguments. (A.2) Let V be a compact subset of Rr . Then, there exists a constant K such that f (t, x, u) ≤ K 1 + |x| , for all (t, x, u) ∈ [0, ∞) × Rn × V, where | · | denotes the usual Euclidean norm. Define

U = ν = [v1 , . . . , vr ] ∈ Rr : αi ≤ vi ≤ βi , i = 1, . . . , r ,

(2)

where αi , i = 1, . . . , r, and βi , i = 1, . . . , r, are given real numbers. A piecewise continuous function u is said to be an admissible control iff u(t) ∈ U , ∀t ∈ [0, T ]. Let U be the class of all such admissible controls. Furthermore, let x(·|u) denote the solution of system (1a)–(1b) corresponding to u ∈ U . Consider the continuous state inequality constraints, given by gi t, x(t|u), u(t) ≤ 0, ∀t ∈ [0, T ], i = 1, . . . , N. (3) It is assumed that the following condition be satisfied. (A.3) gi , i = 1, . . . , N , are continuously differentiable with respect to all its arguments. Now we state the problem as follows. Problem (P) Given the dynamical system (1a)–(1b), find a control u ∈ U such that the cost function T J (u) = Φ0 x(T ) + (4) L0 t, x(t), u(t) dt 0

is minimized subject to the terminal state constraint (1c) and the continuous state and control inequality constraints (3). We assume that the following conditions be satisfied. (A.4) Φ0 is continuously differentiable with respect to x. (A.5) L0 is continuously differentiable with respect to all its arguments. Remark 2.1 By (A.1) and the definition of U , it follows from an argument similar to that given for the proof of Lemma 6.4.2 in [11] that there exists a compact subset X ⊂ Rn such that x(t|u) ∈ X ∀t ∈ [0, T ] and ∀u ∈ U .

264


3 Control Parametrization To solve Problem (P), we shall apply the control parametrization scheme [11] together with a time scaling transform [10]. It is briefly reviewed below. The time horizon [0, T ] is partitioned with a sequence τ = {τ0 , . . . , τp } of time points τi , i = 1, . . . , p − 1. Then, the control is approximated by a piecewise constant function as follows: p u (t|σ, τ ) = σ i χ[τi−1 ,τi ) (t), p

(5)

i=1

where τi−1 ≤ τi , i = 1, . . . , p, with τ0 = 0 and τp = T , and 1, if t ∈ I, χI (t) = 0, otherwise. As up ∈ U , σ i = [σ1i , . . . , σri ] ∈ U for i = 1, . . . , p. Denote by Ξ the set of all such σ = [(σ 1 ) , . . . , (σ p ) ] ∈ Rpr . The switching times τi , 1 ≤ i ≤ p − 1, are also regarded as decision variables. We shall employ the time scaling transform introduced in [10] to map these switching times into fixed time points pk , k = 1, . . . , p − 1, on a new time horizon [0, 1]. This is easily achieved by the differential equation dt (s) = υ p (s), ds

s ∈ [0, 1],

(6a)

with initial condition t (0) = 0,

(6b)

where υ p (s) =

p

θi χ[ i−1 , i ) (s). p

i=1

p

(7)

Here, θi ≥ 0, i = 1, . . . , p. Let θ = [θ1 , . . . , θp ] and let Θ be the set containing all such θ . Taking integration of (6a) with initial condition (6b), it is easy to see that, for k s ∈ [ k−1 p , p ), k = 1, . . . , p, t (s) =

k−1 θk i=1

p

+

θk (ps − k + 1), p

(8)

where k = 1, . . . , p. Clearly, for k = 1, . . . , p − 1, τk =

k θi i=1

p

,

(9)


265

and t (1) =

p θi i=1

p

= T.

(10)

Let Θ˜ be a subset of Θ such that (10) is satisfied. The approximate control given by (5) in the new time horizon [0, 1] becomes i u˜ p (s) = up t (s) = σ χ[ i−1 , i ) (s), p

p

i=1

p

(11)

which has fixed switching times at s = p1 , . . . , p−1 p . Now, by using the time scaling transform (6a)–(6b), the dynamic system (1a)–(1b) is transformed into dy(s) = θk f t (s), y(s), σ k , ds dt (s) = υ p (s), ds y(0) = x 0 ,

s ∈ Jk , k = 1, . . . , p,

(12a) (12b)

t (0) = 0,

(12c)

and the terminal conditions (1c) and (10) become y(1) = x f , respectively, where y(s) = x(t (s)) and ⎧ k−1 k , , ⎪ ⎪ ⎨ p p k−1 k Jk = p ,p , ⎪ ⎪ ⎩ k−1 , k , p

p

t (1) = T ,

(12d)

k = 1, k ∈ {2, . . . , p − 1}, k = p.

We then rewrite system (12a)–(12c) as dy(s) ˜ = f˜ s, y(s), ˜ σ, θ , ds

s ∈ [0, 1],

y(0) ˜ = y˜ 0 ,

(13a) (13b)

with the terminal conditions y(1) ˜ = y˜ f ,

(13c)

where y(s) , t (s) , p k k=1 θk f (t (s), y(s), σ )χJk (s) ˜ f s, y(s), ˜ σ, θ = , υ p (s)

y(s) ˜ =

(14) (15)

266


x0 , 0 f x y˜ f = . T

y˜ 0 =

(16) (17)

To proceed further, let y(·|σ, ˜ θ ) denote the solution of system (13a)–(13b) corresponding to (σ, θ ) ∈ Ξ × Θ. Remark 3.1 As in Remark 2.1, there exists a compact subset Y ⊂ Rn+1 such that ˜ y(s|σ, ˜ θ ) ∈ Y ∀s ∈ [0, 1] and (σ, θ ) ∈ Ξ × Θ. Similarly, applying the time scaling transform to the continuous inequality constraints (3) and the cost function (4) yields gi t (s|θ ), y(s|σ, ˜ θ ), σ k ≤ 0,

s ∈ Jk , k = 1, . . . , p, i = 1, . . . , N,

(18)

and J˜(σ, θ ) = Φ0 y(1|σ, θ ) +

L¯ 0 s, y(s|σ, ˜ θ ), σ, θ ds,

(19)

L¯ 0 s, y(s|σ, ˜ θ ), σ, θ = υ p (s)L0 t (s), x t (s) , u˜ p (s) .

(20)

1

0

respectively, where

Remark 3.2 By respective assumptions specified in (A.1), (A.2) and (A.5), it follows from Remark 3.1 that there exits constants K1 > 0 and K2 > 0 such that Υ s, y(s|σ, ˜ ˜ θ ), σ, θ ≤ K1 , s ∈ Jk , k = 1, . . . , p, (σ, θ ) ∈ Ξ × Θ; ∂Υ (s, y(s|σ, ˜ θ ), σ, θ ) ˜ ≤ K2 , s ∈ Jk , k = 1, . . . , p, (σ, θ ) ∈ Ξ × Θ; ∂σ ∂Υ (s, y(s|σ, ˜ θ ), σ, θ ) ˜ ≤ K2 , s ∈ Jk , k = 1, . . . , p, (σ, θ ) ∈ Ξ × Θ; ∂θ ∂Υ (s, y(s|σ, ˜ θ ), σ, θ ) ˜ ≤ K2 , s ∈ Jk , k = 1, . . . , p, (σ, θ ) ∈ Ξ × Θ, ∂ y˜ where Υ is used to denote f˜i , i = 1, . . . , n, gi (t (s|θ ), y(s|σ, ˜ θ ), σ k ), i = 1, . . . , N, ¯ k = 1, . . . , p, and L0 . The approximate problem to Problem (P) may now be stated formally as follows. Problem (P(p)) Given system (12a)–(12c), find a (σ, θ ) ∈ Ξ × Θ such that the cost function (19) is minimized subject to (12d) and (18).


267

4 An Exact Penalty Function Method Problem (P(p)) is an optimization problem subject to both the equality constraints (12d) and the continuous inequality constraints (18). To solve this problem, an exact penalty function methods introduced in [9] and [12] are used. First, we define F = (σ, θ, ) ∈ Ξ × Θ × R+ : gi t (s|θ ), y(s|σ, θ ), σ k ≤ γ Wi , ∀s ∈ Jk , k = 1, . . . , p, i = 1, . . . , N ,

(21)

where R+ = {α ∈ R : α ≥ 0}, Wi ∈ (0, 1), i = 1, . . . , N , are fixed constants and γ is a positive real number. In particular, when = 0, let F0 = (σ, θ ) ∈ Ξ × Θ : gi t (s|θ ), y(s|σ, θ ), σ k ≤ 0, ∀s ∈ Jk , k = 1, . . . , p; i = 1, . . . , N . Similarly, we define = (σ, θ, ) ∈ F : y(1|σ, ˜ θ ) − y˜ f = 0 ,

(22)

and 0 = (σ, θ ) ∈ F0 : y(1|σ, ˜ θ ) − y˜ f = 0 . Clearly, Problem (P(p)) is equivalent to the following problem, which is denoted ˆ as Problem (P(p)). Given system (13a)–(13b), find a (σ, θ ) ∈ 0 such that the cost function (19) is minimized. Then, by applying a new exact penalty functions introduced in [9] and [12], we obtain a new cost function defined below. J˜δ (σ, θ, ) ⎧ J˜(σ, θ ), ⎪ ⎪ ⎪ ⎪ ⎨ = ⎪ ⎪ ⎪ J˜(σ, θ ) + −α ((σ, θ, ) + 1 ) + δ β , ⎪ ⎩ +∞,

if = 0, gi (t (s|θ ), y(s|θ, ˜ σ ), σ k ) ≤ 0 (23) (s ∈ Jk , k = 1, . . . , p), if > 0, otherwise.

Here, δ > 0 is a penalty parameter, (σ, θ, ), which is referred to as the continuous inequality constraint violation, is defined by (σ, θ, ) =

N i=1 0

1

2 max 0, g¯ i s, y(s|σ, ˜ θ ), σ − γ Wi ds,

(24)

268


where ˜ σ = gi t (s), y(s), u˜ p (s) g¯ i s, y(s), =

p gi t (s), y(s), σ k χJk (s),

(25)

k=1

and u˜ p is defined by (11). Furthermore, 1 , which is referred to as the equality constraint violation, is defined by 2 ˜ θ ) − y˜ f 1 = y(1|σ, =

n+1 f 2 y˜i (1|σ, θ ) − y˜i ,

(26) (27)

i=1

where | · | denotes the usual Euclidean norm, α and γ are positive real numbers, and β > 2. Remark 4.1 Note that other types of equality constraints, such as interior point constraints (see [11]) can be dealt with similarly by introducing appropriate equality constraint violation as defined by (26). We now introduce a surrogate optimal parameter selection problem, which is referred to as Problem (Pδ (p)), as follows. Given system (13a)–(13b), find a (σ, θ, ) ∈ Ξ × Θ × [0, +∞) such that the cost function (23) is minimized. Intuitively, during the process of minimizing J˜δ (σ, θ, ), if σ is increased, β should be reduced. This means that should be reduced as β is fixed. Thus −α will be increased, and hence the constraint violation will be reduced. This means that the values of N

1

2 max 0, g¯ i s, y(s|σ, ˜ θ ), σ − γ Wi ds,

i=1 0

and n+1 f 2 y˜i (1|σ, θ ) − y˜i , i=1

must go down. In this way, the satisfaction of the continuous inequality constraints (18) and the equality constraints (12d) will eventually be achieved. Before presenting the gradient formulas of the cost function of Problem (Pδ (p)), we will rewrite the cost function in the canonical form as in [11] below.


269

J˜δ (σ, θ, ) = Φ0 y(1|σ, θ ) + + −α

1

L¯ 0 s, y(s|σ, ˜ θ ), σ, θ ds

0 N 1

2 max 0, g¯ i s, y(s|σ, ˜ θ ), σ − γ Wi ds

i=1 0

n+1 f 2 + y˜i (1|σ, θ ) − y˜i + δ β

i=1

n+1 f 2 ˜ θ ) + −α y˜i (1|σ, θ ) − y˜i + δ β = Φ0 y(1|σ,

i=1

1

+

L¯ 0 s, y(s|σ, ˜ θ ), σ, θ ds

0

+

−α

N

1

2 ds. max 0, g¯ i s, y(s|σ, ˜ θ ), σ − γ Wi

(28)

i=1 0

Let n+1 f 2 ˜ θ ), = Φ0 y(1|σ, θ ) + −α y˜i (1|σ, θ ) − y˜i + δ β , Φ˜ 0 y(1|σ,

(29)

i=1

and L˜ 0 s, y(s|σ, ˜ θ ), σ, θ, ˜ θ ), σ, θ = L¯ 0 s, y(s|σ, N 1 2 max 0, g¯ i s, y(s|σ, ˜ θ ), σ − γ Wi ds. + −α

(30)

i=1 0

We then substitute (29) and (30) into (28) to give 1 L˜ 0 s, y(s|σ, J˜δ (σ, θ, ) = Φ˜ 0 y(1|σ, ˜ θ ), + ˜ θ ), σ, θ, ds.

(31)

0

Now, the cost function of Problem (Pδ (p)) is in canonical form. As derived for the proof of Theorem 5.2.1 in [11], the gradient formulas of the cost function (31) are given in the following theorem. Theorem 4.1 The gradients of the cost function J˜δ (σ, θ, ) with respect to σ , θ , and are 1 ∂ J˜δ (σ, θ, ) ∂H0 (s, y(s|σ, ˜ θ ), σ, θ, , λ0 (s|σ, θ, )) = ds, (32) ∂σ ∂σ 0 1 ∂H0 (s, y(s|σ, ˜ θ ), σ, θ, , λ0 (s|σ, θ, )) ∂ J˜δ (σ, θ, ) = ds, (33) ∂θ ∂θ 0

270


∂ J˜δ (σ, θ, ) ∂ = −α

−α−1

N

1


i=1 0 n+1 f 2 y˜i (1|σ, θ ) − y˜i +

i=1

− 2γ

γ −α−1

=

1

˜ θ ), σ − γ Wi Wi ds + δβ β−1 max 0, g¯ i s, y(s|σ,

i=1 0

−α−1

N

−α

N

1


i=1 0

+ 2γ

N

1

˜ θ ), σ − γ Wi − γ Wi ds max 0, g¯ i s, y(s|σ,

i=1 0

n+1 f 2 y˜i (1|σ, θ ) − y˜i −α + δβ β−1 ,

(34)

i=1

respectively, where H0 (s, y(s|σ, θ ), σ, θ, , λ(s)) is the Hamiltonian function for the cost function (19) given by ˜ θ ), σ, θ, , λ(s|σ, θ, ) H0 s, y(s|σ, = L˜ 0 s, y(s|σ, ˜ θ ), σ, θ, + λ0 (s|σ, θ, ) f˜ s, y(s|σ, ˜ θ ), σ, θ , (35) and λ0 (·|σ, θ, ) is the solution of the following system of co-state differential equations: ∂H0 (s, y(s|σ, (dλ0 (s)) ˜ θ ), σ, θ, , λ0 (s)) =− , ds ∂ y˜

(36a)

with the boundary condition ∂ Φ˜ 0 (y(1|σ, ˜ θ ), ) . λ0 (1) = ∂ y˜

(36b)

Remark 4.2 By (A.1)–(A.5), Remark 3.1 and Remark 3.2, it follows from arguments similar to those given for the proof of Lemma 6.4.2 in [11] that there exits a compact set Z ⊂ Rn such that λ0 (s|σ, θ, ) ∈ Z, ∀s ∈ [0, 1], (σ, θ ) ∈ Ξ × Θ˜ and ≥ 0. 5 Convergence Analysis Our main aim is to show that, under some mild assumptions, if the parameter δk is sufficient large (δk → +∞ as k → +∞) and (σ (k),∗ , θ (k),∗ , (k),∗ ) is a local min-


271

imizer of Problem (Pδk (p)), then (k),∗ → ∗ = 0, and (σ (k),∗ , θ (k),∗ ) → (σ ∗ , θ ∗ ) with (σ ∗ , θ ∗ ) being a local minimizer of Problem (P(p)). For every positive integer k, let (σ (k),∗ , θ (k),∗ ) be a local minimizer of Problem (Pδk (p)). To obtain our main result, we need Lemma 5.1 Let (σ (k),∗ , θ (k),∗ , (k),∗ ) be a local minimizer of Problem (Pδk (p)). Suppose that J˜δk (σ (k),∗ , θ (k),∗ , (k),∗ ) is finite and that (k),∗ > 0. Then (k),∗ (k),∗ (k),∗ σ ∈ / k , ,θ , where k is defined by (22). Proof Since (σ (k),∗ , θ (k),∗ , (k),∗ ) is a local minimizer of Problem (Pδk (p)) and (k),∗ > 0, we have ∂ J˜δk (σ (k),∗ , θ (k),∗ , (k),∗ ) = 0. ∂

(37)

On a contrary, we assume that the conclusion of the lemma is false. Then, we have γ (k),∗ ≤ (k),∗ Wi , gi y˜ s|σ (k),∗ , θ (k),∗ , σ j ∀s ∈ Jj , j = 1, . . . , p, i = 1, . . . , N, and

y˜ 1|σ (k),∗ , θ (k),∗ − y˜ f = 0.

(38)

(39)

Thus, by (38), (39), (34) and (37), we obtain 0=

∂ J˜δk (σ (k),∗ , θ (k),∗ , (k),∗ ) = βδk β−1 > 0. ∂

This is a contradiction, and hence completing the proof.

Before we introduce the definition of the constraint qualification, we first define f φi y(1|σ, ˜ θ ) = y˜i (1|σ, θ ) − y˜i ,

i = 1, . . . , n + 1.

(40)

Definition 5.1 It is said that the constraint qualification is satisfied for the continuous inequality constraints (18) at (σ, θ ) = (σ¯ , θ¯ ) if the following implication is valid. Suppose that

N 1 0

1

αi (s)

˜ σ¯ , θ¯ ), σ¯ ) ∂ g¯ i (s, y(s| ds = 0. ∂σ

Then, αi (s) = 0, ∀s ∈ [0, 1], i = 1, . . . , N .

272


Theorem 5.1 Suppose that (σ (k),∗ , θ (k),∗ , (k),∗ ) is a local minimizer of Problem (Pδk (p)) such that J˜δk (σ (k),∗ , θ (k),∗ , (k),∗ ) is finite and (k),∗ > 0. If (σ (k),∗ , θ (k),∗ , (k),∗ ) → (σ ∗ , θ ∗ , ∗ ) as k → +∞, and the constraint qualification is satisfied for the continuous inequality constraints (18) at (σ, θ ) = (σ ∗ , θ ∗ ), then ∗ = 0 and (σ ∗ , θ ∗ ) ∈ 0 . Proof From Lemma 5.1, it follows that (σ (k),∗ , θ (k),∗ , (k),∗ ) ∈ / (k),∗ . Now, we construct J˜δ (σ, θ, ) ⎧ J˜(σ, θ ), ⎪ ⎪ ⎨ = ˜ ˆ ⎪ θ, ) + 1 ) + δ β , J (σ, θ ) + −α ((σ, ⎪ ⎩ +∞,

if = 0, g¯ i (s, y(s|σ, ˜ θ ), σ ) ≤ 0 (s ∈ [0, 1]), if > 0, otherwise,

where ˆ (σ, θ, ) =

N 1

2 max 0, g¯ i s, y(s|σ, ˜ θ ), σ − γ Wi ds.

0 i=1

Thus, we have ∂ J˜δ (σ (k),∗ , θ (k),∗ , (k),∗ ) ∂σ 1 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ , (k),∗ , λ (s|σ (k),∗ , θ (k),∗ , (k),∗ )) ∂H0 (s, y(s|σ ˜ 0 ds = ∂σ 0 1 ¯ (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ L0 (s, y(s|σ ˜ ds = ∂σ 0 N −α 1 + 2 (k),∗ max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ 0 i=1 (k),∗ , θ (k),∗ ), σ (k),∗ ) γ ∂ g¯ i (s, y(s|σ ˜ − (k),∗ Wi ds ∂σ 1 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) (k),∗ (k),∗ (k),∗ ∂ f˜(s, y(s|σ ˜ + ds λ0 s|σ ,θ , ∂σ 0

= 0,

(41)

∂ J˜δ (σ (k),∗ , θ (k),∗ , (k),∗ ) ∂ (k),∗ −α−1 = 1 N γ 2 max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ − (k),∗ Wi × −α ds 0 i=1


+ 2γ

N 1

273

γ max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ − (k),∗ Wi

0 i=1

γ × − (k),∗ Wi ds

n+1 (k),∗ (k),∗ β−1 f 2 y˜i 1|σ − y˜i ,θ −α + δk β (k),∗ i=1

= 0.

(42)

Suppose that (k),∗ → ∗ = 0. Then, by (42), it can be shown by using Remark 3.1 and Remark 3.2 and Lebesgue dominated convergence theorem that its first term tends to a finite value, while the last term tends to infinity as δk → +∞, when k → +∞. This is impossible for the validity of (42). Thus, ∗ = 0. Now, by (41), we have

1 0

(k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ L¯ 0 (s, y(s|σ ˜ ds ∂σ N −α 1 γ + 2 (k),∗ max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ − (k),∗ Wi 0 i=1 (k),∗ , θ (k),∗ ), σ (k),∗ ) ∂ g¯ i (s, y(s|σ ˜ ds ∂σ 1 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) (k),∗ (k),∗ (k),∗ ∂ f˜(y(s|σ ˜ + ds λ0 s|σ ,θ , ∂σ 0

×

= 0. Thus, lim

k→+∞

0

1

(k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ L¯ 0 (s, y(s|σ ˜ ds ∂σ

−α + 2 (k),∗

N 1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

0 i=1 (k),∗ , θ (k),∗ ), σ (k),∗ ) γ ∂ g¯ i (s, y(s|σ ˜ − (k),∗ Wi ds ∂σ 1 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) (k),∗ (k),∗ (k),∗ ∂ f˜(s, y(s|σ ˜ + ds λ0 s|σ ,θ , ∂σ 0

= 0. By Remarks 3.2 and 4.1, it follows from the Lebesgue dominated convergence theorem that the first and third terms converge to some finite values. On the other hand,

274


the second term tends to infinite, which is impossible, and hence

N

1

lim

0 k→+∞

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1

(k),∗ , θ (k),∗ ), σ (k),∗ ) ˜ ∂ g¯ i (s, y(s|σ × ds ∂σ =

N

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

×

(k),∗ , θ (k),∗ ), σ (k),∗ ) ∂ g¯ i (s, y(s|σ ˜ ds ∂σ

= 0.

(43)

Since the constraint qualification is satisfied for the continuous inequality constraints (18) at (σ, θ ) = (σ ∗ , θ ∗ ), it follows that, for each i = 1, . . . , N , ∗ ∗ max 0, g¯ i s, y(s|σ , θ ), σ ∗ = 0, ˜ for each s ∈ [0, 1]. This, in turn, implies that, for each i = 1, . . . , N , ∗ ∗ g¯ i s, y(s|σ , θ ), σ ∗ ≤ 0, ˜

(44)

for each s ∈ [0, 1]. Next, from (44) and (42), it is easy to see that, for each i = 1, . . . , n + 1, when k → +∞, y˜i (1|σ ∗ , θ ∗ ) − y˜i = 0. f

The proof is completed.

(45)

We have the following corollary. Corollary 5.1 Suppose that (σ (k),∗ , θ (k),∗ ) → (σ ∗ , θ ∗ ) ∈ 0 and that (k),∗ → ∗ = 0. Then, (σ (k),∗ , θ (k),∗ , (k),∗ ) → (σ ∗ , θ ∗ , ∗ ) = 0, and 1 → 0. In what follows, we shall turn our attention to the exact penalty function constructed in (23). We shall see that, under some mild conditions, J˜δ (σ, θ, ) is continuously differentiable. We assume that the following conditions be satisfied. (A.6) ξ max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ = o (k),∗ , ξ > 0, s ∈ [0, 1], i = 1, 2, . . . , N.

(46)


275

(A.7) ξ , φi y˜ 1|σ (k),∗ , θ (k),∗ = o (k),∗

ξ > 0, i = 1, . . . , n + 1.

(47)

Theorem 5.2 Suppose that γ > α, ξ > α, ξ > α, −α − 1 + 2ξ > 0, −α − 1 + 2ξ > 0, 2γ − α − 1 > 0. Then, J˜δk σ (k),∗ , θ (k),∗ , (k),∗ (k),∗ → ∗ =0 −−−−−−−−−−−−−−−−→ J˜δk σ (k),∗ , θ (k),∗ , 0 = J˜(σ ∗ , θ ∗ ), (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(48)

∇(σ,θ,) J˜δk σ (k),∗ , θ (k),∗ , (k),∗ (k),∗ → ∗ =0 −−−−−−−−−−−−−−−−→ ∇(σ,θ,) J˜δk (σ ∗ , θ ∗ , 0) = ∇(σ,θ) J˜(σ ∗ , θ ∗ ), 0 . (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(49)

Proof Based on the conditions of the theorem, we can show that, for = 0, lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

=

J˜δk σ (k),∗ , θ (k),∗ , (k),∗

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

N −α + (k),∗

J˜ σ (k),∗ , θ (k),∗

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ 2 ds − (k),∗ Wi n+1 (k),∗ (k),∗ (k),∗ β (k),∗ −α f 2 y˜i 1|σ − y˜i + δk ,θ + .

(50)

i=1

By an argument similar to that given for the proof of Lemma 6.4.3 in [11], we can show that when (σ (k),∗ , θ (k),∗ ) → (σ ∗ , θ ∗ ), ∗ ∗ , θ ), ˜ y˜ s|σ (k),∗ , θ (k),∗ → y(s|σ

(51)

∀s ∈ [0, 1]. By (51) and (19), it follows from an argument similar to that given for the proof of Lemma 6.4.4 in [11] that lim

(σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

Substituting (52) into (50), we have

J˜ σ (k),∗ , θ (k),∗ = J˜(σ ∗ , θ ∗ ).

(52)

276


lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

J˜δk σ (k),∗ , θ (k),∗ , (k),∗

= J˜(σ ∗ , θ ∗ ) +

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

N 1 max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ i=1 0

(k),∗ α −1 (k),∗ γ 2 Wi ds − n+1 +

(k),∗ , θ (k),∗ ) − y˜ f )2 i . ( (k),∗ )α

i=1 (y˜i (1|σ

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(53)

For the second term of (53), we have N 1 max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ lim (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

i=1 0

(k),∗ γ 2 α −1 − Wi ds × (k),∗

=

lim

N 1

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

− α max 0, (k),∗ 2 g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ − α 2 2W ds. − (k),∗ i

Since ξ > α, γ > α, it follows from (A.6) that, for any s ∈ [0, 1], − α lim max 0, (k),∗ 2 g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

γ − α 2W − (k),∗ i = 0.

(54)

Thus, by Remark 3.2 and the Lebesgue dominated convergence theorem, we obtain N 1 − α lim max 0, (k),∗ 2 g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

i=1 0

γ − α 2 2W ds − (k),∗ i N 1 = lim i=1 0

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

− α max 0, (k),∗ 2 g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

γ − α 2 2W ds − (k),∗ i = 0.

(55)


277

Similarly, for the third term of (53), it is clear from (A.7) that n+1

(k),∗ , θ (k),∗ ) − y˜ f )2 i ( (k),∗ )α

i=1 (y˜i (1|σ

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

= 0.

Combining (53), (55) and (56) gives lim J˜δk σ (k),∗ , θ (k),∗ , (k),∗ = J˜δk (σ ∗ , θ ∗ , 0) = J˜(σ ∗ , θ ∗ ). (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(56)

(57)

For the second part of the theorem, we need the gradient formulas of J˜(σ, θ ), which can be derived in the same way as that given for the proof of Theorem 5.2.1 in [11]. These gradient formulas are given by ∂ J˜(σ, θ ) = ∂σ

∂ H¯ 0 (s, y(s|σ, ˜ θ ), σ, θ, λ¯ 0 (s|σ, θ )) ds, ∂σ

(58)

1 ¯ ∂ J˜(σ, θ ) ∂ H0 (s, y(s|σ, ˜ θ ), σ, θ, λ¯ 0 (s|σ, θ )) = ds, ∂θ ∂θ 0

(59)

0

1

¯ where H¯ 0 (s, y(s|σ, θ ), σ, θ, λ(s|σ, θ )) is the Hamiltonian function defined by ¯ H¯ 0 s, y(s|σ, ˜ θ ), σ, θ, λ(s|σ, θ) = L¯ 0 s, y(s|σ, ˜ θ ), σ, θ + λ¯ 0 (s|σ, θ ) f˜ s, y(s|σ, ˜ θ ), σ, θ ,

(60)

and λ¯ 0 (·|σ, θ ) is the solution of the following system of co-state differential equations: ∂ H¯ 0 (s, y(s|σ, ˜ θ ), σ, θ, λ¯ 0 (s)) (dλ¯ 0 (s)) =− , (61a) ds ∂ y˜ with the boundary condition ∂Φ0 (y(1|σ, θ )) . λ¯ 0 (1) = ∂y

(61b)

By (60), we can rewrite (61a) as ∂ f˜(s, y(s|σ, (dλ¯ 0 (s)) ∂ L¯ 0 (s, y(s|σ, ˜ θ ), σ, θ ) ˜ θ ), σ, θ ) =− − λ¯ 0 (s) . ds ∂ y˜ ∂ y˜

(62)

By (62) with terminal condition (61b) and (36a) with terminal condition (36b), we obtain (k),∗ (k),∗ λ¯ 0 s|σ − λ0 s|σ (k),∗ , θ (k),∗ , (k),∗ ,θ = ¯λ0 1|σ (k),∗ , θ (k),∗ − λ0 1|σ (k),∗ , θ (k),∗ , (k),∗

278


0 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ L¯ 0 (ω, y(ω|σ ˜ − + ∂ y˜ 1 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ˜ ∂ L˜ 0 (ω, y(ω|σ dω + ∂ y˜ s ˜ (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ f (s, y(ω|σ ˜ + ∂ y˜ 1 × −λ¯ 0 ω|σ (k),∗ , θ (k),∗ + λ0 ω|σ (k),∗ , θ (k),∗ , (k),∗ dω.

(63)

By the definitions of (36b), (61b), (29) and ξ > α, it follows from (A.7) that lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

=

(k),∗ (k),∗ λ¯ 0 1|σ − λ0 1|σ (k),∗ , θ (k),∗ , (k),∗ ,θ

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

n+1 (k),∗ −α ∂ (k),∗ (k),∗ f 2 y˜i 1|σ − y˜i ,θ ∂y i=1

= 0.

(64)

On the other hand, by (30), ξ > α and γ > α, it follows from (A.6), Remark 3.2 and Lebesgue dominated convergence theorem that, for each s ∈ [0, 1], lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

+ =

0 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ¯ ˜ − ∂ L0 (ω, y(ω|σ ∂ y˜ 1

(k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ , (k),∗ ) ∂ L˜ 0 (ω, y(ω|σ ˜ dω ∂ y˜ 0 −α lim 2 (k),∗

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

×

1

N max 0, g¯ i ω, y˜ ω|σ (k),∗ , θ (k),∗ , σ (k),∗ − γ Wi i=1

×

(k),∗ , θ (k),∗ ), σ (k),∗ ) ∂ g¯ i (ω, y(ω|σ ˜ dω ∂ y˜

= 0.

(65)

Thus, by applying Gronwall–Bellman’s Lemma to (63), it follows from (64), (65), Remark 3.2 and the Lebesgue dominated theorem that, for each s ∈ [0, 1], lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(k),∗ (k),∗ λ¯ 0 s|σ − λ0 s|σ (k),∗ , θ (k),∗ , (k),∗ = 0. ,θ

(66)


279

By (32), (35), and (30), we have lim ∇σ J˜δk σ (k),∗ , θ (k),∗ , (k),∗ (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

=

1

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

N (k),∗ −α +2

0 1

(k),∗ , θ (k),∗ ), σ (k),∗ ) ∂ L¯ 0 (s, y(s|σ ˜ ds ∂σ

γ max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ − (k),∗ Wi

i=1 0 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ g¯ i (s, y(s|σ ˜ ds ∂σ 1 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) (k),∗ (k),∗ (k),∗ ∂ f˜(s, y(s|σ ˜ + ds λ0 s|σ ,θ , ∂σ 0 1 ∂ L¯ (s, y(s|σ (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ˜ 0 = lim ds ∂σ (k),∗ → ∗ =0 0

×

(σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

+ 0

+

(k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ f˜(s, y(s|σ ˜ ds λ0 s|σ (k),∗ , θ (k),∗ , (k),∗ ∂σ N 1 −α lim max 0, (k),∗ 2

1

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

i=1 0

× g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

(k),∗ , θ (k),∗ ), σ (k),∗ ) (k),∗ γ −α ∂ g¯ i (s, y(s|σ ˜ ds . − Wi ∂σ

(#)

Then, by Remark 3.2 and the Lebesgue dominated convergence theorem, it follows from (66) that 1 ∂ L¯ (s, y(s|σ (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ˜ 0 lim ds ∂σ (k),∗ → ∗ =0 0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

+

1

λ0 s|σ (k),∗ , θ (k),∗ , (k),∗

0

(k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ f˜(s, y(s|σ ˜ ds × ∂σ 1 (k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ L¯ 0 (s, y(s|σ ˜ = lim ds ∂σ (k),∗ → ∗ =0 0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

280


+

1

lim

0

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(k),∗ (k),∗ (k),∗ λ0 s|σ ,θ ,

(k),∗ , θ (k),∗ ), σ (k),∗ , θ (k),∗ ) ∂ f˜(s, y(s|σ ˜ ds ∂σ 1 ¯ ∗ , θ ∗ ), σ ∗ , θ ∗ ) ∂ L0 (s, y(s|σ ˜ ds = ∂σ 0 1 ∗ , θ ∗ ), σ ∗ , θ ∗ ) ∂ f˜(s, y(s|σ ˜ + ds λ¯0 (s|σ ∗ , θ ∗ ) ∂σ 0

×

= ∇σ J˜(σ ∗ , θ ∗ ).

(67)

Similarly, by Remark 3.2, (A.6) and ξ > α, γ > α, it follows from the Lebesgue dominated convergence theorem that N 1 −α (k),∗ (k),∗ (k),∗ lim ,σ max 0, (k),∗ g¯ i s, y˜ s|σ ,θ 2 (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

i=1 0

(k),∗ , θ (k),∗ ), σ (k),∗ ) γ −α ∂ g¯ i (s, y(s|σ ˜ ds Wi − (k),∗ ∂σ

=2

N i=1 0

1

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

−α (k),∗ (k),∗ (k),∗ max 0, (k),∗ ,σ g¯ i s, y˜ s|σ ,θ

(k),∗ , θ (k),∗ ), σ (k),∗ ) γ −α ∂ g¯ i (s, y(s|σ ˜ ds Wi − (k),∗ ∂σ = 0.

We substitute (67) and (68) into (#) to give lim ∇σ J˜δk σ (k),∗ , θ (k),∗ , (k),∗ = ∇σ J˜(σ ∗ , θ ∗ ). (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(68)

(69)

Similarly, we can show that lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

∇θ J˜δk σ (k),∗ , θ (k),∗ , (k),∗ = ∇θ J˜(σ ∗ , θ ∗ ).

On the other hand, we note that lim ∇ J˜δk σ (k),∗ , θ (k),∗ , (k),∗ (k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

=

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

(k),∗ −α−1

(70)


× −α

281

N 1 γ 2 max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ − (k),∗ Wi ds i=1 0

+ 2γ

N

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ γ − (k),∗ Wi − (k),∗ Wi ds n+1 (k),∗ (k),∗ 2 (k),∗ β−1 φi y˜ 1|σ ,θ + + σk β i=1

=

lim

(k),∗ → ∗ =0 (σ (k),∗ ,θ (k),∗ )→(σ ∗ ,θ ∗ )∈0

−α

N

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

− α+1 (k),∗ γ − α+1 2 2 − 2 W × (k),∗ i ds + 2γ

N

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ γ −α−1 − (k),∗ Wi − (k),∗ Wi (k),∗ ds n+1 (k),∗ (k),∗ (k),∗ − α+1 2 2 ,θ φi y˜ 1|σ + . i=1

The proof is completed.

Theorem 5.3 Let (k),∗ → ∗ = 0 and (σ (k),∗ , θ (k),∗ ) → (σ ∗ , θ ∗ ) ∈ 0 be such that J˜δk (σ (k),∗ , θ (k),∗ , (k),∗ ) is finite. Then, (σ ∗ , θ ∗ ) is a local minimizer of Problem (P(p)). Proof On a contrary, assume that (σ ∗ , θ ∗ ) be not a local minimizer of Problem (P(p)). Then, there must exist a feasible point (σˆ ∗ , θˆ ∗ ) ∈ N (σ ∗ , θ ∗ ) of Problem (P(p)) such that J˜(σˆ ∗ , θˆ ∗ ) < J˜(σ ∗ , θ ∗ ),

(71)

where N (σ ∗ , θ ∗ ) is a -neighborhood of (σˆ ∗ , θˆ ∗ ) in 0 for some > 0. Since (σ (k),∗ , θ (k),∗ , (k),∗ ) is a local minimizer of Problem (Pδk ), there exists a sequence {ξk }, such that J˜δk σ, θ, (k),∗ ≥ J˜δk σ (k),∗ , θ (k),∗ , (k),∗ , for any (σ, θ ) ∈ Nξk (σ (k),∗ , θ (k),∗ ). Now, we construct a sequence of feasible points {(σˆ (k),∗ , θˆ (k),∗ )} of Problem (P(p)) satisfying (k),∗ (k),∗ (k),∗ (k),∗ ξk ≤ . σˆ − σ , θˆ ,θ k

282


Clearly,

J˜δk σˆ (k),∗ , θˆ (k),∗ , (k),∗ ≥ J˜δk σ (k),∗ , θ (k),∗ , (k),∗ .

(72)

Letting k → +∞, we have lim σˆ (k),∗ , θˆ (k),∗ − (σˆ ∗ , θˆ ∗ ) ≤ lim σˆ (k),∗ , θˆ (k),∗ − σ (k),∗ , θ (k),∗ k→+∞

k→+∞

+ lim σ (k),∗ , θ (k),∗ − (σ ∗ , θ ∗ ) k→+∞

+ (σ ∗ , θ ∗ ) − (σˆ ∗ , θˆ ∗ ) ≤ 0 + 0 + . However, > 0 is arbitrary. Thus, lim σˆ (k),∗ , θˆ (k),∗ = (σˆ ∗ , θˆ ∗ ). k→+∞

(73)

(74)

Letting k → +∞ in (72), it follows from the first part of Theorem 5.2 and (74) that lim J˜δk σˆ (k),∗ , θˆ (k),∗ , (k),∗ k→+∞

= J˜(σˆ ∗ , θˆ ∗ ) ≥ lim J˜δk σ (k),∗ , θ (k),∗ , (k),∗ k→+∞

= J˜(σ ∗ , θ ∗ ). This is a contradiction to (71), and hence it completes the proof.

(75)

Theorem 5.4 Let −α − β + 2ξ > 0, −α − β + 2ξ > 0 and −α − β + 2γ > 0. Then, there exists a k0 > 0, such that (k),∗ = 0, (σ (k),∗ , θ (k),∗ ) is local minimizer of Problem (P(p)), for k ≥ k0 . Proof On a contrary, we assume that the conclusion be false. Then, there exists a subsequence of {(σ (k),∗ , θ (k),∗ , (k),∗ )}, which is denoted by the original sequence, such that, for any k0 > 0, there exists a k > k0 satisfying (k ),∗ = 0. By Theorem 5.1, we have (k),∗ (k),∗ → (σ ∗ , θ ∗ ) ∈ 0 , as k → +∞. σ ,θ (k),∗ → ∗ = 0, Since (k),∗ = 0, ∀k, it follows from dividing (34) by ( (k),∗ )β−1 that N 1 (k),∗ −α−β max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ −α i=1 0

γ 2 − (k),∗ Wi ds + 2γ

N i=1 0

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗


283

γ γ − (k),∗ Wi − (k),∗ Wi ds

n+1 (k),∗ (k),∗ f 2 y˜i 1|σ − y˜i ,θ −α + δk β = 0.

(76)

i=1

This is equivalent to N (k),∗ −α−β −α

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ 2 − (k),∗ Wi ds + 2γ

N

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ γ − (k),∗ Wi − (k),∗ Wi + max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ γ − (k),∗ Wi g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ − max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ γ − (k),∗ Wi g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ ds n+1 (k),∗ (k),∗ f 2 y˜i 1|σ − y˜i ,θ −α + δk β i=1

= 0.

(77)

Rearranging (77) yields N 1 (k),∗ −α−β max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ (2γ − α) i=1 0

n+1 (k),∗ (k),∗ (k),∗ γ 2 f 2 y˜i 1|σ − y˜i Wi ds − α ,θ − + δk β i=1 N −α−β = 2γ (k),∗

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ − (k),∗ Wi g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ ds.

(78)

Note that −α − β + 2ξ > 0 and −α − β + 2ξ > 0. Then, by Remark 3.2 and the Lebesgue dominated convergence theorem, we can show that the left hand side of

284


(78) yields N (k),∗ −α−β (2γ − α)

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ 2 − (k),∗ Wi ds

n+1 (k),∗ (k),∗ f 2 y˜i 1|σ − y˜i ,θ −α + δk β → ∞.

(79)

i=1

However, under the same conditions and −α − β + 2γ > 0, we can show, also by Remark 3.2 and the Lebesgue dominated convergence theorem, that the right hand side of (78) gives N −α−β 2γ (k),∗

1

max 0, g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗

i=1 0

γ − (k),∗ Wi g¯ i s, y˜ s|σ (k),∗ , θ (k),∗ , σ (k),∗ ds → 0. This is a contradiction. Thus, the proof is completed.

(80)

Theorem 5.5 Let up,∗ be an optimal control of the approximate problem (P(p)). Suppose that u∗ be an optimal control of Problem (P). Then, lim J up,∗ = J (u∗ ).

p→+∞

Proof The proof is similar to that given for the proof of Theorem 8.6.2 in [11].

(81)

Theorem 5.6 Let up,∗ be an optimal control of the approximate Problem (P (p)), and u∗ be an optimal control of the problem (P ). Suppose that lim up,∗ = u, ¯

p→+∞

a.e. on [0, T ].

(82)

Then, u¯ is an optimal control of the Problem (P ), and lim J up,∗ = J (u∗ ).

p→+∞

Proof The proof is similar to that given for the proof of Theorem 8.6.3 in [11].

(83)

6 Algorithm With the results in the previous sections, we provide the following algorithm for solving Problem (P).


285

Algorithm 6.1 Step 1. Set δ (1) = 10, (1) = 0.1, ∗ = 10−9 , β > 2, choose an initial point (σ 0 , θ 0 , 0 ), the iteration index k = 0. The values of γ and α are chosen depending on the specific structure of Problem (P) concerned. Step 2. Solve Problem (Pδk ), and let (σ (k),∗ , θ (k),∗ , (k),∗ ) be the minimizer obtained. Step 3. If (k),∗ > ∗ , δ (k) < 108 , set δ (k+1) = 10 × σ (k) , k := k + 1. Go to Step 2 with (σ (k),∗ , θ (k),∗ , (k),∗ ) as the new initial point in the new optimization process. Else set (k),∗ := ∗ , then go to Step 4. Step 4. Check the feasibility of (σ (k),∗ , θ (k),∗ ) (i.e., check whether or not (k),∗ , θ (k),∗ ), σ (k),∗ ) ≤ 0. max max g¯ i (s, y(s|σ ˜ 1≤i≤N s∈[0,1] If (σ (k),∗ , θ (k),∗ )

is feasible, then it is a local minimizer of Problem (P(p)). Exit. Else go to Step 5. Step 5. Adjust the parameters α, β and γ such that the conditions of Lemma 5.1 are satisfied. Set δ (k+1) = 10δ (k) , (k+1) = 0.1 (k) , k := k + 1. Go to Step 2.

Remark 6.1 In Step 3, if (k),∗ > ∗ , it follows from Theorems 5.1 and 5.2 that (σ (k),∗ , θ (k),∗ ) cannot be a feasible point. This means that the penalty parameter δ is not chosen large enough. Thus we need to increase δ. If δk > 108 , but still (k),∗ > ∗ , then we should adjust the value of α, β and γ , such that the conditions of Theorem 5.2 are satisfied. Then, go to Step 2. Remark 6.2 Clearly, we cannot check the feasibility of gi (y(s), σ ) ≤ 0, i = 1, . . . , N , for every s ∈ [0, 1]. In practice, we choose a set which contains a dense enough of points in [0, 1]. Check the feasibility of gi (y(s), σ ) ≤ 0 over this set for each i = 1, . . . , N . Remark 6.3 Although we have proved that a local minimizer of the exact penalty function optimization problem (Pδk ) will converge to a local minimizer of the original problem (P(p)), we need, in actual computation, set a lower bound ∗ = 10−9 for (k),∗ so as to avoid the situation of being divided by (k),∗ = 0, leading to infinity.

7 Examples Example 1 The following optimal control problem is taken from [11] and [13]: min g0 :=

1

2 2 2 x1 (t) + x2 (t) + 0.005 u(t) dt

(84)

0

subject to

·

x 1 (t) = x2 (t),

(85a)

x 2 (t) = −x2 (t) + u(t),

(85b)

·

286


Fig. 1 Optimal state variables for Example 1

Fig. 2 Optimal control and the resulting constraint function for Example 1

with initial conditions x1 (0) = 0,

x2 (0) = −1,

(86)

and the continuous state inequality constraint g1 = 8(t − 0.5)2 − 0.5 − x2 (t) ≥ 0,

∀t ∈ [0, 1],

(87)

together with the control constraints −20 ≤ u(t) ≤ 20,

∀t ∈ [0, 1].

(88)

In this problem, we set p = 20, γ = 3 and W1 = 0.3. The result is shown below. The optimal objective function value is g0∗ = 1.75101803 × 10−1 , where δ = 1.0 × 106 and = 1.89531e × 10−5 . The continuous inequality constraints (87) is satisfied ∀t ∈ [0, 1]. Comparing with the results obtained, for Example 6.7.2 in [11], the minimum value of the objective function is almost the same (it is 0.1730 in [11]). However, in [11], the continuous inequality constraints (87) is slightly violated at some t ∈ [0, 1]. The optimal control, the optimal state and the constraint are shown in Figs. 1 and 2. Example 2 We consider a realistic and complex problem of transferring containers from a ship to a cargo truck at the port of Kobe. It is taken from [14]. The container crane is driven by a hoist motor and a trolley drive motor. For safety reason, the objective is to minimize the swing during and at the end of the transfer. The problem


287

is summarized after appropriate normalization by 1 2 2 x3 (t) + x6 (t) dt , minimize g0 = 4.5

(89)

0

subject to the dynamical equations ⎧· ⎪ ⎪ ⎪x 1 (t) = 9x4 (t), ⎪ ⎪ · ⎪ ⎪ x 2 (t) = 9x5 (t), ⎪ ⎪ ⎪ ⎪ ⎪ · ⎪ ⎪ ⎨x 3 (t) = 9x6 (t) · ⎪ x 4 (t) = 9 u1 (t) + 17.2656x3 (t) , ⎪ ⎪ ⎪ ⎪ ⎪ · ⎪ ⎪ x 5 (t) = 9u2 (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩x· 6 (t) = − 9 u1 (t) + 27.0756x3 (t) + 2x5 (t)x6 (t) , x2 (t)

(90a) (90b) (90c) (90d) (90e) (90f)

where

x(0) = [0, 22, 0, 0, −1, 0] ,

(91a)

(91b)

x(1) = [10, 14, 0, 2.5, 0, 0] , and

u1 (t) ≤ 2.83374, −0.80865 ≤ u2 (t) ≤ 0.71265,

(92) ∀t ∈ [0, 1],

with continuous state inequality constraints x4 (t) ≤ 2.5, ∀t ∈ [0, 1], x5 (t) ≤ 1.0, ∀t ∈ [0, 1].

(93)

(94a) (94b)

The bounds on the states can be formulated as the continuous inequality constraints as follows: g1 = −x4 (t) + 2.5 ≥ 0,

(95)

g2 = x4 (t) + 2.5 ≥ 0,

(96)

g3 = −x5 (t) + 1.0 ≥ 0,

(97)

g4 = x5 (t) + 1.0 ≥ 0.

(98)

In this problem, we set p = 20, γ = 3 and W1 = W2 = W3 = W4 = 0.3. The result obtained is shown below. The optimal objective function value is g0∗ = 5.75921513 × 10−3 , where δ = 1.0 × 105 and = 1.00057 × 10−7 . All the continuous inequality

288



Fig. 4 The optimal state variables for Example 2


constraints are satisfied ∀t ∈ [0, 1]. Comparing with the results obtained, for Example 6.7.3 in [11], our minimum value of the objective function is slightly larger (it is 4.684 × 10−3 in [11]). However, in [11], the continuous inequality constraints are not completely satisfied ∀t ∈ [0, 1]. The optimal state variables, the optimal control and the constraints are shown in Figs. 3–8, respectively.


289

Fig. 6 Optimal controls for Example 2

Fig. 7 The constraints function under the optimal control for Example 2

Fig. 8 The constraints function under the optimal control for Example 2

Example 3 The following problem is taken from [7]: Find a control u : [0, 4.5] → R that minimizes the cost function 4.5 2 u (t) + x12 (t) dt, (99) 0

subject to the dynamical equations

·

x 1 (t) = x2 (t), · x 2 (t) = −x1 (t) + x2 (t)(1.4 − 0.14x22 (t)) + 4u(t),

(100a) (100b)

with the initial conditions

x1 (0) = −5,

(101a)

x2 (0) = −5,

(101b)

290



Fig. 10 Optimal control and the resulting constraint function for Example 3

and the continuous inequality constraint 1 g1 = −u(t) − x1 (t) ≥ 0, 6

t ∈ [0, 4.5].

(102)

In this problem, we set p = 10, γ = 3 and W1 = 0.3. The result is shown below. The optimal objective function value obtained is g0∗ = 4.58048380e × 101 , where δ = 1.0 × 104 and = 9.99998 × 10−5 . The continuous inequality constraint (102) is satisfied ∀t ∈ [0, 4.5]. In [3], the optimal objective function value is about 4.6961921e × 101 , which is slightly larger than our result. The optimal state variables, the optimal control and the constraint are shown in Figs. 9 and 10.

8 Conclusions This paper presents a new exact penalty function method for continuous inequality constrained optimal control problems. It shows that, for a sufficiently large penalty parameter value any local minimizer of the transformed problem is a local minimizer of the original problem. From results obtained for the three examples, we see that the method proposed is effective. In particular, the optimal controls obtained are feasible controls.


291

Acknowledgements This work was supported by China Scholarship Council, a grant from the Australia Research Council and the Innovation Team Program of the National Science Foundation of China under Grant No. 61021002 Robust flight control theory and applications.

References 1. Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995) 2. Teo, K.L., Jennings, L.S.: Nonlinear optimal control problems with continuous state inequality constraints. J. Optim. Theory Appl. 63(1), 1–22 (1989) 3. Loxton, R.C., Teo, K.L., Rehbock, V., Yiu, K.F.C.: Optimal control problems with a continuous inequality constraint on the state and control. Automatica 45, 2250–2257 (2009) 4. Buskens, C., Maurer, H.: SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real time control. J. Comput. Appl. Math. 120(1– 2), 85–108 (2000) 5. Chen, T.W.C., Vassiliadis, V.S.: Inequality path constraints in optimal control: a finite iteration ε-convergent scheme based on pointwise discretization. J. Process Control 15(3), 353–362 (2005) 6. Huyer, W., Neumaier, A.: A new exact penalty function. SIAM J. Optim. 13(4), 1141–1158 (2003) 7. Gerdts, M.: Global convergence of a nonsmooth Newton method for control-state constrained optimal control problems. SIAM J. Control Optim. 19(1), 326–350 (2008) 8. Gerdts, M.: A nonsmooth Newton’s method for control-state constrained optimal control problems. Math. Comput. Simul. 79(4), 925–936 (2008) 9. Gerdts, M., Kunkel, M.: A nonsmooth Newton’s method for discretized optimal control problems with state and control constraints. J. Ind. Manage. Optim. 4(2), 247–270 (2008) 10. Teo, K.L., Jennings, L.S., Lee, H.W.J., Rehbock, V.L.: The control parameterization enhancing transformation for constrained optimal control problems. J. Aust. Math. Soc. Ser. B, Appl. Math 40, 314– 335 (1997) 11. Teo, K.L., Goh, C.J., Wong, K.H.: A Unified Computational Approach for Optimal Control Problems. Longman, New York (1991) 12. Yu, C., Teo, K.L., Zhang, L., Bai, Y.: A new exact penalty function method for continuous inequality constrained optimization problems. J. Ind. Manag. Optim. 6(4), 895–910 (2010) 13. Mehra, R.K., Davis, R.E.: A generalized gradient method for optimal control problems with inequality constraints and singular Arcs. IEEE Trans. Autom. Control AC-17, 69–78 (1972) 14. Sakawa, Y., Shindo, Y.: Optimal control of container cranes. Automatica 18, 257–266 (1982)