Optimal control for problems with a general variational inequality

by Helmut Dietrich Martin-Luther-University Halle-Wittenberg Institute of Optimization and Stochastics 06099 Halle Germany

Abstract We study first order optimality conditions for the control of a system by a general variational inequality under use of a fitting penalty method. For grasp the general variational inequality we define and determine a regularized gap function and generate a minimizing sequence of the control problem by means of Ekeland’s variational principle. The optimality system we obtain by passing to the limit. Key Words : general variational inequality, gap function, Ekeland’s variational principle, penalty method, necessary optimality condition Mathematics Subject Classification 1991 : Primary : 49J52, 49J40 Secondary : 90C26

1 Introduction and statement of the problem

In this paper we investigate optimal control problems governed by general variational inequalities. The notion of general variational inequality is introduced 1988 by Noor [16]. Such problems arise frequently for example in shape optimization, in connection with obstacle problems, in game theory, in economic modelling and in design of transportation networks so that their study is of an important practical interest. Compare the following papers Ben-El-Mechaiekh and Isac [4], Xui, Zhang and Noor [21], Xui and Zhang [23] and Noor [13] . The control problem as an optimization problem is also refered to as generalized bilevel programming problem or mathematical program with equilibrium constraints, see for instance [9], [22]. Already for problems of optimal control with variational inequalities it is known that one cannot get in general a classical optimality system, see Mignot [10], Mignot and Puel [11], Barbu [2], [3], Shi [20], Wenbin and Rubio [19], Bergounioux and Dietrich [5]. There exist several possibilities to derive necessary optimality conditions for such problems. Mignon [10] used the notion of the conical derivative, on the other hand one can formulate the variational inequality as an operator equation with a set-valued operator, see for example Barbu [2], [3], Outrata [17], and apply the results of nonsmooth and set-valued analysis. Furthermore, one can construct a so-called gap function to a variational inequality, so that the variational inequality is equivalent to a scalar equation of the gap function. Under suitable assumptions such a gap function is Fréchet differentiable, compare the regularized gap function for a variational inequality of Fukushima [7], and the state of the art about variational and quasivariational inequalities in 1995 by Giannessi and Maugheri [8]. See also Baiocci and Capelo [1]. Then one may use a penalty method to approximate the optimal control problem with a variational inequality, see for example Shi [20], Wenbin and Rubio [19], Bergounioux and Dietrich [5], Dietrich [6]. For our control problem we also use a gap function for grasp the general variational inequality. We define and calculate a regularized gap function in the sense of Fukushima [7] to the general variational inequality and determine their Fréchet derivative. Following we approximate by means of a suitable penalty function the control problem (P) and generate a minimizing sequence by Ekeland’s variational principle. The limit process leads to a necessary optimality condition of (P).

Consider the following abstract problem of optimal control with a general variational inequality

(P)  inf (P) : = inf_{y,u} F (y, u)
subject to (y, u) ∈ L with

L := \{(y, u) ∈ B_{1} × B_{2} | \begin{matrix}find(y, u) ∈ Y × U : g(y, u) ∈ K, so that \\ ⟨T (y, u), g(x, v) − g(y, u)⟩ ≥ 0 \\ ∀(x, v) ∈ Y × U : g(x, v) ∈ K, \end{matrix} \}

under the following assumptions, summarized by (H) and are always fulfilled in the sequel :

• B_{1}, B_{2} are reflexive Banach spaces with the dual spaces B_{1}^{∗}, B_{2}^{∗}, the duality mappings Λ_{i} : B_{i} → B_{i}^{∗} and the duality pairings ⟨· , ·⟩_{i} , i = 1, 2. B_{1} is the space of the state and the nonempty convex and closed set Y ⊂ B_{1} is the set of state constraints for the problem. Similar is B_{2} the space of control and the nonempty convex and closed set U ⊂ B_{2} is the set of control constraints. Finally, H is a Hilbert space with the dual space H^{∗}, the duality pairing ⟨· , ·⟩ , with the scalar product (· , ·) and the duality mapping Λ : H → H^{∗}, i.e. the canonical isomorphism between H and H^{∗}.

• The set M_{1} with Y ×U ⊂ M_{1} ⊂ B_{1}×B_{2} is assumed as an open set in B_{1}×B_{2}. The mappings T : M_{1} ⊂ B_{1} × B_{2} → H^{∗} and g : M_{1} ⊂ B_{1} × B_{2} → H are in general nonlinear and continuous operators. At last is K ⊂ H a nonempty set.

• F : Y ×U ⊂ B_{1} ×B_{2} → ℝ is a convex and closed function bounded from below.

• There exists an optimal solution (y_{0}, u_{0}) ∈ L of (P).

2 The gap function

To the general variational inequality above we define a so-called regularized gap function

h_{α}(y, u) = sup_{(x,v)∈Y ×U,g(x,v)∈K} [−α ⟨T (y, u), g(x, v) − g(y, u)⟩

−\frac{1}{2} ∥g(x, v) − g(y, u)∥^{2}] , α > 0.  (2.1)

The gap function (2.1) is a generalization of Fukushima’s regularized gap function [7] for variational inequalities adapted to general variational inequalities. For feasible elements of the general variational inequality the gap function (2.1) possesses nonnegative values :

(y, u) ∈ Y × U and g(y, u) ∈ K ⟹ h_{α}(y, u) ≥^{(x,v)=(y,u)} 0,  (2.2)

If (y, u) is a solution of the general variational inequality

(y, u) ∈ Y × U and g(y, u) ∈ K : ⟨T (y, u), g(x, v) − g(y, u)⟩ ≥ 0
∀(x, v) ∈ Y × U : g(x, v) ∈ K,

then we get

0 ≥ −α ⟨T (y, u), g(x, v) − g(y, u)⟩ − \frac{1}{2} ∥g(x, v) − g(y, u)∥^{2}
∀(x, v) ∈ Y × U : g(x, v) ∈ K,

0 ≥ h_{α}(y, u) ≥^{(x,v)=(y,u)} 0

and finally

h_{α}(y, u) = 0.

With the condition

K̃ := \overline{g(Y × U)} ∩ K ⊂ H is a convex and closed set in H,  (2.3)

it follows on the other hand from

(y, u) ∈ Y × U and g(y, u) ∈ K : h_{α}(y, u) = 0

the inequality

0 ≥ −α ⟨T (y, u), g(x, v) − g(y, u)⟩ − \frac{1}{2} ∥g(x, v) − g(y, u)∥^{2}
∀(x, v) ∈ Y × U and g(x, v) ∈ K,

and with

g(x, v) = tg(z, w) + (1 − t)g(y, u) g(z, w) ∈ K̃, t ∈ (0, 1]

we have

α ⟨T (y, u), g(z, w) − g(y, u)⟩ ≥ −\frac{t}{2} ∥g(z, w) − g(y, u)∥^{2} as t → +0

and finally

⟨T (y, u), g(z, w) − g(y, u)⟩ ≥ 0 ∀(z, w) ∈ Y × U and g(z, w) ∈ K.

Combined one received the following

Theorem 2.1 If K̃ = \overline{g(Y × U)} ∩ K ⊂ H is a convex and closed set in H, then the following equivalence

\begin{matrix}Find (y, u) ∈ Y × U and g(y, u) ∈ K : \\ ⟨T (y, u), g(x, v) − g(y, u)⟩ ≥ 0 \\ ∀(x, v) ∈ Y × U : g(x, v) ∈ K \end{matrix} \} ⟺ \{ \begin{matrix}(y, u) ∈ Y × U and \\ g(y, u) ∈ K : \\ h_{α}(y, u) = 0, \end{matrix} \} ⟺

g(y, u) = PK̃ (g(y, u) − αΛ^{−1}T (y, u))  (2.4)

is valid.

Here, PK̃(·) denotes the projection in the Hilbert space H onto the closed and convex set K̃ ⊂ H. For the gap function we get ( K̃ ⊂ H is convex and closed, α > 0 )

h_{α}(y, u) = sup_{(x,v)∈Y ×U,g(x,v)∈K} [−α ⟨T (y, u), g(x, v) − g(y, u)⟩ − \frac{1}{2} ∥g(x, v) − g(y, u)∥^{2}]
= \frac{α^{2}}{2} ∥T (y, u)∥_{∗}^{2} − \frac{1}{2} ∥(I − PK̃) (g(y, u) − αΛ^{−1}T (y, u))∥^{2} .  (2.5)

One must calculate the supremum of a strongly concave function on the convex and closed set K̃. Consequently there exists an unique solution, namely the projection yK̃ := PK̃ (g(y, u) − αΛ^{−1}T (y, u)). If we additional assume that the operators

T : M_{1} ⊂ B_{1} × B_{2} → H^{∗} and g : M_{1} ⊂ B_{1} × B_{2} → H  (2.6)

are C^{1} differentiable in the Fréchet sense on the open set M_{1} ⊂ B_{1} × B_{2} , then the gap function above is also Fréchet differentiable with the following derivative on M_{1} , see Phelps [18],

h′_{α}(y, u) = α^{2} [T′(y, u)]^{∗} Λ^{−1}T (y, u)
− ([g′(y, u)]^{∗} − α [T′(y, u)]^{∗} Λ^{−1}) Λ (I − PK̃) (g(y, u) − αΛ^{−1}T (y, u)) .  (2.7)

This implies the following connection at a point (y_{1}, u_{1})

h′_{α}(y_{1}, u_{1}) = α · [g′(y_{1}, u_{1})]^{∗}T (y_{1}, u_{1}),  (2.8)

which is a solution of the general variational inequality, i.e. for (y_{1}, u_{1}) with h_{α}(y_{1}, u_{1}) = 0. Especially for the optimal point of (P) we have

h′_{α}(y_{0}, u_{0}) = α · [g′(y_{0}, u_{0})]^{∗}T (y_{0}, u_{0}).

Example 2.2 Consider a simple problem of optimal control which is used for the explanation of the means.

min(P1) : = min \{y^{2} + u^{2} | \begin{matrix}(y + u − 1)(x + v^{2} − y − u^{2}) ≥ 0 \\ ∀(x, v) ∈ ℝ^{2} : x + v^{2} ≥ 1 \\ (y, u) ∈ ℝ^{2} : y + u^{2} ≥ 1 \end{matrix} \}
T (y, u) = y + u − 1, g(y, u) = y + u^{2}, K = [1, +∞), K̃ = \overline{g(ℝ^{2})} ∩ K = K

First, we solve the general variational inequality problem

Case 1 : T (y, u) = y + u − 1 = 0
⟹ L_{1} = {(y, u) = (1 − u, u) ∈ ℝ^{2} | u ∈ (−∞, 0] ∪ [1, +∞)}
Case 2 : T (y, u) = y + u − 1 > 0
⟹ L_{2} = {(y, u) = (1 − u^{2}, u) ∈ ℝ^{2} | u ∈ (0, 1)}
Case 3 : T (y, u) = y + u − 1 < 0
⟹ L_{3} = ∅
L = \{(y, u) ∈ ℝ^{2} | \begin{matrix}y = 1 − u for u ∈ (−∞, 0] ∪ [1, +∞) \\ y = 1 − u^{2} for u ∈ (0, 1) \end{matrix} \} .

We obtain as the unique optimal solution of (P1) (u_{opt}, y_{opt}) = (\frac{1}{2} \sqrt{2}, \frac{1}{2}) with min(P1) = \frac{3}{4}.

Next, we calculate the gap function of the general variational inequality problem

h_{1}(y, u) = \frac{1}{2} (y + u − 1)^{2} − \frac{1}{2} [(I − P_{[1,+∞)}) (u^{2} − u + 1)]^{2}
= \{ \begin{matrix}\frac{1}{2} (y + u − 1)^{2} − \frac{1}{2} (u^{2} − u)^{2} for u ∈ (0, 1) \\ \frac{1}{2} (y + u − 1)^{2} for u ∈ (−∞, 0] ∪ [1, +∞) \end{matrix} .

One can immediately prove that the equivalence

(y, u) ∈ ℝ^{2} : y + u^{2} ≥ 1 and h_{1}(y, u) = 0 ⟺ (y, u) ∈ L

is valid.

3 The penalty method in the general case

First, apply theorem 2.1 and obtain as an equivalent problem to (P)

(P_{α}) inf (P_{α}) := inf \{F (y, u) | \begin{matrix}h_{α}(y, u) = 0, g(y, u) ∈ K \\ y ∈ Y, u ∈ U \end{matrix} \} ,

where α > 0 is an arbitrary parameter. The problem (P_{α}) is an optimization problem with a smooth equality and an inclusion-type constraint. But it is not possible in general to apply the Lagrange method since the regularity condition is not fulfilled. The set \{(y, u) ∈ Y × U | g(y, u) ∈ K \} may be nonconvex, merely K̃ = \overline{g(Y × U)}∩K is by the assumption convex and closed, see the example above. Additionally suppose in the sequel that the set K ⊂ H is also convex and closed. We approximate the problem (P_{α}) by

(P_{α}^{n}) inf {F_{n}(y, u) | (y, u) ∈ Y × U }

with

F_{n}(y, u) = F (y, u)+ \frac{n}{2} ·h_{α}^{2}(y, u)+ \frac{n}{2} ∥(I − P_{K}) (g(y, u))∥^{2} + \frac{1}{2} ∥y − y_{0}∥_{1}^{2} + \frac{1}{2} ∥u − u_{0}∥_{2}^{2} ,

where (y_{0}, u_{0}) ∈ Y × U : g(y_{0}, u_{0}) ∈ K is an optimal solution of (P). The cost function F_{n} is finite and lower semicontinuous on the convex and closed set Y × U ⊂ B_{1} × B_{2}. The quadratic terms

\frac{1}{2}∥y − y_{0}∥^{2} and \frac{1}{2}∥u − u_{0}∥^{2}

have the effect to obtain the strong convergence of the minimizing sequence in the limit process under determined conditions, see Barbu [2], [3]. By Ekeland’s variational principle there exists for each ε_{n} > 0, n = 1, 2, ... with ε_{n} → +0 an element (y_{n}, u_{n}) ∈ Y × U such that

F_{n}(y_{n}, u_{n}) ≤ inf(P_{α}^{n}) + ε_{n}

and

F_{n}(y, u) ≥ F_{n}(y_{n}, u_{n}) − ε_{n} ∥( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )∥_{1×2} ∀(y, u) ∈ Y × U.  (3.1)

Put (y, u) = (y_{0}, u_{0}) ∈ Y × U̇ : g(y_{0}, u_{0}) ∈ K into the inequality of Ekeland’s variational principle, it follows

F (y_{0}, u_{0}) − F (y_{n}, u_{n}) + ε_{n} · ∥( \begin{matrix}y_{0} − y_{n} \\ u_{0} − u_{n} \end{matrix} )∥_{1×2} ≥ \frac{n}{2} · h_{α}^{2}(y_{n}, u_{n})

+\frac{n}{2} ∥(I − P_{K})(g(y_{n}, u_{n})∥^{2} + \frac{1}{2} ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}^{2} ≥ 0.  (3.2)

If the cost function is Gâteaux differentiable at the optimal point (y_{0}, u_{0}), the last inequality implies

(∥F ′(y_{0}, u_{0})∥ + ε_{n}) · ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2} ≥ \frac{n}{2} · h_{α}^{2}(y_{n}, u_{n})

+\frac{n}{2} ∥(I − P_{K}) (g(y_{n}, u_{n}))∥^{2} + \frac{1}{2} ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}^{2} ≥ 0.  (3.3)

It follows that the sequence {(y_{n}, u_{n})} is bounded. There exist weakly convergent subsequences of {y_{n}} ⊂ Y ⊂ B_{1} and {u_{n}} ⊂ U ⊂ B_{2} , again still denoted by {y_{n}} and {u_{n}} , such that

y_{n} ⇀ ȳ ∈ Y, u_{n} ⇀ ū ∈ U .

Furthermore it follows from (3.3) that

h_{α}(y_{n}, u_{n}) → 0 and ∥g(y_{n}, u_{n}) − P_{K}(g(y_{n}, u_{n}))∥ → 0 as n → +∞.  (3.4)

If the operator g : M_{1} ⊂ B_{1} × B_{2} → H is sequentially weakly-strongly continuous, then

(y_{n}, u_{n}) ⇀ (ȳ, ū) ⟹ g(y_{n}, u_{n}) → g(ȳ, ū) ∈ K.  (3.5)

For the limit element (ȳ, ū) ∈ Y × U : g(ȳ, ū) ∈ K is valid. Now we prove that h_{α}(ȳ, ū) = 0, if the operator T : M_{1} ⊂ B_{1} × B_{2} → H^{∗} is additional also sequentially weakly-strongly continuous. By

y_{n} ⇀ ȳ, u_{n} ⇀ ū ⟹

yn ; K̃ := PK̃(g(y_{n}, u_{n}) − αΛ^{−1}T (y_{n}, u_{n})) → ȳK̃ := PK̃ (g(ȳ, ū) − αΛ^{−1}T (ȳ, ū)) ,

we obtain

h_{α}(y_{n}, u_{n}) → h_{α}(ȳ, ū) = 0.(⟹ ȳK̃ = g(ȳ, ū) )

Therefore, the element (ȳ, ū) is a feasible element of (P) . From the inequality (3.2) we get as n → +∞

0 ≥ F (y_{0}, u_{0}) − F (ȳ, ū) ≥ F (y_{0}, u_{0}) − lim inf_{n→+∞} F (y_{n}, u_{n}) ≥

lim sup_{n→+∞} [\frac{n}{2} · h_{α}^{2}(y_{n}, u_{n}) + \frac{n}{2} ∥(I − P_{K}) (g(y_{n}, u_{n}))∥^{2} + \frac{1}{2} ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}^{2}] ≥ 0.

(3.6)

With it we obtain the following result, which is the basic of the limit process to get the optimality system.

Theorem 3.1 Suppose (H) , K ⊂ H and K̃ = \overline{g(Y × U)}∩K are convex and closed, the cost function F : Y × U ⊂ B_{1} × B_{2} → ℝ is Gâteaux differentiable at the optimal solution (y_{0}, u_{0}) of (P) and the operators g, T are weakly- strongly continuous. Then, a subsequence {(y_{n}, u_{n})} generated by Ekeland’s variational principle strongly converges to (y_{0}, u_{0}) and furthermore we get

lim_{n→+∞} \frac{n}{2} · h_{α}^{2}(y_{n}, u_{n}) = 0, and  (3.7)
lim_{n→+∞} \frac{n}{2} ∥(I − P_{K}) (g(y_{n}, u_{n}))∥^{2} = 0.  (3.8)

4 The penalty method in a special case

Now we consider the following special case, that additionally the set

M = {(y, u) ∈ Y × U ⊂ B_{1} × B_{2} | g(y, u) ∈ K}  (4.1)

is convex and closed. Then one can simpler approximate the problem (P_{α}) by (P̃_{α}^{n}).

(P̃_{α}^{n}) inf{F̃_{n}(y, u) | (y, u) ∈ M }

with

F̃_{n}(y, u) = F (y, u) + n · h_{α}(y, u) + \frac{1}{2}∥y − y_{0}∥_{1}^{2} + \frac{1}{2}∥u − u_{0}|_{2}^{2}.

How above we apply Ekeland’s variational principle and obtain a minimizing sequence (y_{n}, u_{n}) ∈ M, so that the following inequality holds, corresponding to (3.2)

F (y_{0}, u_{0}) − F (y_{n}, u_{n}) + ε_{n} · ∥( \begin{matrix}y_{0} − y_{n} \\ u_{0} − u_{n} \end{matrix} )∥_{1×2} ≥ n · h_{α}(y_{n}, u_{n})

+\frac{1}{2} ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}^{2} ≥ 0.  (4.2)

If the cost function is again Gâteaux differentiable at the optimal point (y_{0}, u_{0}), this implies the following inequality belonging to (3.3)

(∥F ′(y_{0}, u_{0})∥ + ε_{n}) · ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2} ≥ n · h_{α}(y_{n}, u_{n})

+\frac{1}{2} ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}^{2} ≥ 0.  (4.3)

The sequence {(y_{n}, u_{n})} ⊂ M must again be bounded, a subsequence is weakly convergent

M ∋ (y_{n}, u_{n}) ⇀^{B_{1}×B_{2}} (ȳ, ū) ∈ M,

and

lim_{n→+∞} h_{α}(y_{n}, u_{n}) = 0.  (4.4)

If the operators T and g are sequentially weakly-strongly continuous, it follows that

(y_{n}, u_{n}) ⇀ (ȳ, ū) ⇒ h_{α}(y_{n}, u_{n}) → h_{α}(ȳ, ū) = 0.

The element (ȳ, ū) is therefore feasible of (P) we get from (4.2)

0 ≥ F (y_{0}, u_{0}) − F (ȳ, ū) ≥ F (y_{0}, u_{0}) − lim inf_{n→+∞} F (y_{n}, u_{n}) ≥

lim sup_{n→+∞} [n · h_{α}(y_{n}, u_{n}) + \frac{1}{2} ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}^{2}] ≥ 0.  (4.5)

Combined we obtain for this special case Theorem 4.1 Suppose the assumptions of theorem 3.1 and additional that the set M ⊂ B_{1} × B_{2} is convex and closed. Then a subsequence {(y_{n}, u_{n})} generated by Ekeland’s variational principle strongly converges to (y_{0}, u_{0}) and

lim_{n→+∞} n · h_{α}(y_{n}, u_{n}) = 0.  (4.6)

Example 4.2 Consider the following problem of optimal control with a general variational inequality of this special type.

min(P2) : = min \{(y − 3)^{2} + (u + \frac{7}{4})^{2} | \begin{matrix}(y + u)(x^{2} + v − y^{2} − u) ≥ 0 \\ ∀(x, v) ∈ ℝ^{2} : x^{2} + v ≤ 0 \\ (y, u) ∈ ℝ^{2} : y^{2} + u ≤ 0 \end{matrix} \}
T (y, u) = y + u, g(y, u) = y^{2} + u, K = (−∞, 0], K̃ = \overline{g(ℝ^{2})} ∩ K = K.

Obviously, the set M = {(y, u) ∈ ℝ^{2} | g(y, u) = y^{2} + u ≤ 0 } is convex and closed. For the gap function we obtain with α = 1 by (2.5)

h_{1}(y, u) = \{ \begin{matrix}\frac{1}{2}(y + u)^{2} − \frac{1}{2}(y^{2} − y)^{2} & for & y ∈ (−∞, 0) ∪ (1, +∞) \\ \frac{1}{2}(y + u)^{2} & for & y ∈ [0, 1] \end{matrix}

The set of solutions of the variational inequality is

L = \{(y, u) ∈ ℝ^{2} | \begin{matrix}u = −y^{2} & for & y ∈ (−∞, 0) ∪ (1, +∞) \\ u = −y & for & y ∈ [0, 1] \end{matrix} \}

Therefore the optimal solution of the control problem is the unique determined point (y_{0}, u_{0}) = (\frac{3}{2}, −\frac{9}{4}) and min(P2) = \frac{5}{2}.

5 The optimality system in the general case

To derive the optimality system for (P) first we consider the general case and set

y = y_{n} + tr, r ∈ Y − y_{n}, u = u_{n} + ts, s ∈ U − u_{n}, t > 0

into (3.1) and obtain

F_{n}(y_{n} + tr, u_{n} + ts) ≥ F_{n}(y_{n}, u_{n}) − ε_{n} · t · ∥( \begin{matrix}r \\ s \end{matrix} )∥_{1×2}

F (y_{n} + tr, u_{n} + ts) − F (y_{n}, u_{n}) + \frac{n}{2} [h_{α}^{2}(y_{n} + tr, u_{n} + ts) − h_{α}^{2}(y_{n}, u_{n})]

+\frac{n}{2} [∥(I − P_{K}) (g(y_{n} + tr, u_{n} + ts))∥^{2} − ∥(I − P_{K})(g(y_{n}, u_{n}))∥^{2}] +

+\frac{1}{2} ∥( \begin{matrix}y_{n} + tr − y_{0} \\ u_{n} + ts − u_{0} \end{matrix} )∥_{1×2}^{2} − \frac{1}{2} ∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}^{2} + ε_{n} · t · ∥( \begin{matrix}r \\ s \end{matrix} )∥_{1×2} ≥ 0

⟨F ′(y_{n}, u_{n}), ( \begin{matrix}r \\ s \end{matrix} )⟩_{1×2} + n · h_{α}(y_{n}, u_{n}) · ⟨h ′_{α}(y_{n}, u_{n}), ( \begin{matrix}r \\ s \end{matrix} )⟩_{1×2}

+ ⟨( \begin{matrix}Λ_{1}(y_{n} − y_{0}) \\ Λ_{2}(u_{n} − u_{0}) \end{matrix} ) , ( \begin{matrix}r \\ s \end{matrix} )⟩_{1×2}

n · ⟨[g′(y_{n}, u_{n})]^{∗} Λ (I − P_{K}) (g(y_{n}, u_{n}) , ( \begin{matrix}r \\ s \end{matrix} )⟩_{1×2} + ε_{n} · ∥( \begin{matrix}r \\ s \end{matrix} )∥_{1×2} ≥ 0

⟨F ′(y_{n}, u_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + n · h_{α}(y_{n}, u_{n}) · ⟨h ′_{α}(y_{n}, u_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2}

+ ⟨( \begin{matrix}Λ_{1}(y_{n} − y_{0}) \\ Λ_{2}(u_{n} − u_{0}) \end{matrix} ) , ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + ε_{n} · ∥( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )∥_{1×2}

n · ⟨[g′(y_{n}, u_{n})]^{∗} Λ (I − P_{K}) (g(y_{n}, u_{n})), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} ≥ 0.

Now we use (2.7) under the assumption (2.6) and obtain

⟨F ′(y_{n}, u_{n}) + ( \begin{matrix}Λ_{1}(y_{n} − y_{0}) \\ Λ_{2}(u_{n} − u_{0}) \end{matrix} ) , ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + ε_{n} · ∥( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )∥_{1×2}

n · h_{α}(y_{n}, u_{n}) · \{⟨α^{2} [T′(y_{n}, u_{n})]^{∗} Λ^{−1}T (y_{n}, u_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2}

− ⟨([g′(y_{n}, u_{n})]^{∗} − α [T′(y_{n}, u_{n})]^{∗} Λ^{−1}) Λ (I − PK̃) (g(y_{n}, u_{n}) − αΛ^{−1}T (y_{n}, u_{n})) ,

( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2}\} + n · ⟨[g′(y_{n}, u_{n})]^{∗} Λ (I − P_{K}) (g(y_{n}, u_{n})), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} ≥ 0.

We set

p_{n} := n · h_{α}(y_{n}, u_{n}) · [g(y_{n}, u_{n}) − yn ; K̃] ∈ H,  (5.1)
q_{n}^{∗} := n · Λ(I − P_{K})(g(y_{n}, u_{n})) ∈ H^{∗},  (5.2)

and get the following inequality

⟨F ′(y_{n}, u_{n}) + ( \begin{matrix}Λ_{1}(y_{n} − y_{0}) \\ Λ_{2}(u_{n} − u_{0}) \end{matrix} ) , ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + ε_{n} · ∥( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )∥_{1×2}

+α · ⟨[T ′(y_{n}, u_{n})]^{∗} p_{n}, ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + ⟨[g′(y_{n}, u_{n})]^{∗} (q_{n}^{∗} − Λp_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2}

+α · n · h_{α}(y_{n}, u_{n}) · ⟨T (y_{n}, u_{n}), g′(y_{n}, u_{n}) ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩ ≥ 0,

(5.3)

for all (y, u) ∈ Y × U. We summarize the obtained result.

Corollary 5.1 Suppose the assumptions of theorem 3.1 and additional (2.6) as well as (2.7). Then, for the subsequence {(y_{n}, u_{n})} generated by Ekeland’s variational principle it follows the inequality (5.3).

Theorem 5.2 Suppose the assumptions of theorem 3.1.

a) If there exists a constant K_{1} > 0 such that the constraint

lim sup ; \begin{matrix}(y, u) → (y_{0}, u_{0}) \\ (y, u) ∈ Y × U : g(y, u) ∉ K \end{matrix} \frac{|F (y, u) − F (y_{0}, u_{0})|}{∥(I − P_{K})(g(y, u))∥} < K_{1}  (5.4)

is valid, then the sequence {q_{n}^{∗}} ⊂ H^{∗} by (5.2) is bounded and there exists a weakly^{∗} convergent subsequence(again denoted by {q_{n}^{∗}})with q_{n}^{∗} ⇀^{∗} q^{∗} ∈ H^{∗}. b)If there exists a constant K_{2} > 0 such that the inequality

lim sup ; \begin{matrix}(y, u) → (y_{0}, u_{0}) \\ (y, u) ∈ Y × U : h_{α}(y, u) ≠ 0 \end{matrix} |\frac{F (y, u) − F (y_{0}, u_{0})}{h_{α}(y, u)}| · ∥g(y, u) − yK̃∥ < K_{2} (5.5)

holds, then the sequence {p_{n}} ⊂ H is bounded and there exists a weakly convergent subsequence (again denoted by {p_{n}}) with p_{n} ⇀ p ∈ H.

Proof :

a) Consider (3.2), set specifically ϵ_{n} = \frac{1}{n} → +0, then it follows the inequality

F (y_{0}, u_{0}) − F (y_{n}, u_{n}) + \frac{1}{n} ∥( \begin{matrix}y_{0} − y_{n} \\ u_{0} − u_{n} \end{matrix} )∥ ≥ \frac{n}{2}∥(I − P_{K})(g(y_{n}, u_{n}))∥^{2},

and further

\frac{F (y_{0}, u_{0}) − F (y_{n}, u_{n})}{∥(I − P_{K})(g(y_{n}, u_{n}))∥} + \frac{∥( \begin{matrix}y_{0} − y_{n} \\ u_{0} − u_{n} \end{matrix} )∥}{n · ∥(I − P_{K})(g(y_{n}, u_{n}))∥} ≥ \frac{n}{2} · ∥(I − P_{K})(g(y_{n}, u_{n}))∥.

From the last inequality one recognizes that the expression q_{n}^{∗} = n·Λ(I−P_{K})(g(y_{n}, u_{n})) ∈ H^{∗} cannot be unrestricted.

b) Now consider again (3.2), set specifically ϵ_{n} = \frac{1}{n} → +0, then it follows the inequality

F (y_{0}, u_{0}) − F (y_{n}, u_{n}) + \frac{1}{n} ∥( \begin{matrix}y_{0} − y_{n} \\ u_{0} − u_{n} \end{matrix} )∥ ≥ \frac{n}{2} · h_{α}^{2}(y_{n}, u_{n}),

and further

|\frac{F (y_{n}, u_{n}) − F (y_{0}, u_{0})}{h_{α}(y_{n}, u_{n})}| + \frac{∥( \begin{matrix}y_{0} − y_{n} \\ u_{0} − u_{n} \end{matrix} )∥}{n · |h_{α}(y_{n}, u_{n})|} ≥ \frac{n}{2}|h_{α}(y_{n}, u_{n})|.

Multiplying the last inequality with the expression ∥g(y_{n}, u_{n}) − yn ; K̃∥, then we can see by (5.5), that the term p_{n} = n · h_{α}(y_{n}, u_{n}) · [g(y_{n}, u_{n}) − yn ; K̃] is bounded. □ The introduction of the sizes p_{n}, and q_{n}^{∗} and the proof of their boundedness allows the limit process n → +∞ in the inequality (5.3). The weak limits p and q^{∗} of subsequences of {p_{n}} and {q_{n}^{∗}} are the adjoint states - a sort of Lagrange multipliers - of the problem.

Corollary 5.3 Suppose the assumptions of theorem 3.1 and additional (2.6) as well as (2.7). Then from (3.7) it follows that the so-called adjoint equation

⟨T (y_{0}, u_{0}), p⟩ = 0  (5.6)

for the optimal solution (y_{0}, u_{0}) of (P) is valid. Furthermore, from (3.8) we get the variational inequality

⟨q^{∗}, k − g(y_{0}, u_{0})⟩ ≤ 0 ∀k ∈ K.  (5.7)

as an optimality condition for the optimal solution (y_{0}, u_{0}) of the optimal control problem (P).

Proof : From (3.7) it follows for n → +∞ with the abbreviation

yn ; K̃ = PK̃ (g(y_{n}, u_{n} − αΛ^{−1}T (y_{n}, u_{n}))

\frac{n}{2}h_{α}(y_{n}, u_{n}) [−α⟨T (y_{n}, u_{n}), yn ; K̃ − g(y_{n}, u_{n})⟩ − \frac{1}{2}∥yn ; K̃ − g(y_{n}, u_{n})∥^{2}] → 0
\frac{α}{2}⟨T (y_{n}, u_{n}), p_{n}⟩ + \frac{1}{4}⟨Λ(yn ; K̃ − g(y_{n}, u_{n})), p_{n}⟩ → 0

and by theorem 5.2 the validity of the adjoint equation (5.6). On the other hand one get by definition of the projection on the convex and closed set K ⊂ H

(g(y_{n}, u_{n}) − P_{K}(g(y_{n}, u_{n}), k − P_{K}(g(y_{n}, u_{n}))) ≤ 0 ∀k ∈ K.

It follows that the inequality

⟨q_{n}^{∗}, k − P_{K}(g(y_{n}, u_{n}))⟩ ≤ 0 ∀k ∈ K

is valid and by passing to the limit n → +∞ , take into account theorem 5.2, we obtain the condition (5.7). □ Introduce the set-valued mapping A : B_{1} × B_{2} ⇝ B_{1} × B_{2} by the definition

A(y, u) :=

\{ \begin{matrix}\{(ỹ, ũ) ∈ Y × U | h_{α}(y, u) · ⟨A, g′(y, u) ( \begin{matrix}ỹ − y \\ ũ − u \end{matrix} )⟩ ≤ 0 \} & \begin{matrix}for (y, u) \\ ∈ Y × U \end{matrix} \\ ∅  else \end{matrix} (5.8)

with the abbreviation A := αT (y, u) + Λ(yK̃ − g(y, u)). We summarize several simple properties of the set-valued mapping A:

• DomA = Y × U ⊂ B_{1} × B_{2} is nonempty convex and closed,

• A(y, u) = Y × U for all (y, u) ∈ Y × U : h_{α}(y, u) = 0,

• (y, u) ∈ A(y, u) ∀(y, u) ∈ Y × U,

• The multifunction A has convex and closed images.

Theorem 5.4 Suppose the assumptions of theorem 3.1, (2.6), (2.7), (5.4) and (5.5), then there exist two adjoint states p ∈ H and q^{∗} ∈ H^{∗} such that the following optimality conditions for the optimal solution (y_{0}, u_{0}) of (P)

⟨T (y_{0}, u_{0}), p⟩ = 0  (5.9)
⟨q^{∗}, k − g(y_{0}, u_{0})⟩ ≤ 0 ∀k ∈ K (5.10)
⟨F′(y_{0}, u_{0}) + [T′(y_{0}, u_{0})]^{∗}p + [g′(y_{0}, u_{0})]^{∗}q^{∗}, ( \begin{matrix}y − y_{0} \\ u − u_{0} \end{matrix} )⟩ ≥ 0
∀(y, u) ∈ lim sup_{n→+∞} A(y_{n}, u_{n}) ⊂ Y × U  (5.11)

hold.

Proof : By (5.3) it follows for (y, u) ∈ A(y_{n}, u_{n}) and n → +∞ the condition (5.11). Jointly with (5.6) and (5.7) we get the desired optimality system of theorem 5.4. □ We check the optimality system (5.9), (5.10) and (5.11) for the case of example (P1). From (5.9) it follows that T (y_{0}, u_{0}) = \frac{1}{\sqrt{2}} − \frac{1}{2} < 0 ⇒ p = 0, the second condition (5.10) implies q^{∗}(k − g(y_{0}, u_{0})) = q^{∗}(k − 1) ≤ 0 ∀k ∈ K = [1, +∞) also q^{∗} ≤ 0. Finally, the condition (5.11) is valid for q^{∗} = −1. The exact calculation of the set lim sup_{n→+∞} A(y_{n}, u_{n}) is generally impossible since the minimizing sequence (y_{n}, u_{n}) of Ekeland’s variational principle is unknown.

6 The optimality system in the special case

Consider now the special case, that the set

M = {(y, u) ∈ Y × U ⊂ B_{1} × B_{2} | g(y, u) ∈ K}

additionally is convex and closed. From the inequality

F̃_{n}(y, u) ≥ F̃_{n}(y_{n}, u_{n}) − ε_{n} ∥( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )∥_{1×2} ∀(y, u) ∈ M

we conclude as above in the general case

⟨F′(y_{n}, u_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + n · ⟨h′_{α}(y_{n}, u_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} +

⟨( \begin{matrix}Λ_{1}(y_{n} − y_{0}) \\ Λ_{2}(u_{n} − u_{0}) \end{matrix} ) , ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} +ε_{n} ·∥ \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} ∥_{1×2} ≥ 0 ∀(y, u) ∈ M. (6.1)

By (2.7) it follows that

⟨F′(y_{n}, u_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + nα · ⟨[T′(y_{n}, u_{n})]^{∗}(g(y_{n}, u_{n}) − yn ; K̃), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2}

−n · ⟨Λ[g(y_{n}, u_{n}) − yn ; K̃ − αΛ^{−1}T (y_{n}, u_{n})], g′(y_{n}, u_{n}) ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩

+ ⟨( \begin{matrix}Λ_{1}(y_{n} − y_{0}) \\ Λ_{2}(u_{n} − u_{0}) \end{matrix} ) , ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + ε_{n} · ∥( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )∥_{1×2} ≥ 0 ∀(y, u) ∈ M.

(6.2)

Denote by q_{n} := n[g(y_{n}, u_{n}) − yn ; K̃], then we can prove that {q_{n}} ⊂ H is a bounded sequence under suitable conditions.

Theorem 6.1 Suppose the assumptions of theorem 4.1, then the inequality (6.2) is fulfilled for a subsequence {(y_{n}, u_{n})} generated by Ekeland’s variational principle. If the property

∃ K > 0 : lim sup ; \begin{matrix}(y, u) →^{M} (y_{0}, u_{0}) : \\ h_{α}(y, u) ≠ 0 \end{matrix} \frac{|F (y, u) − F (y_{0}, u_{0})|}{∥g(y.u) − yK̃∥} ≤ K  (6.3)

holds, then the sequence {q_{n}} is bounded and there exists a weakly convergent subsequence( again denoted by {q_{n}} )

q_{n} = n[g(y_{n}, u_{n}) − yn ; K̃] ⇀ q̄ ∈ H.

Proof : From the definition of the projection

yn ; K̃ = PK̃(g(y_{n}, u_{n}) − αΛ^{−1}T (y_{n}, u_{n}))

we conclude for z = g(y_{n}, u_{n}) ∈ K̃

(g(y_{n}, u_{n}) − αΛ^{−1}T (y_{n}, u_{n}) − yn ; K̃, g(y_{n}, u_{n}) − yn ; K̃) ≤ 0

and furthermore

\frac{n}{2}∥g(y_{n}, u_{n}) − yn ; K̃∥^{2} ≤ n · h_{α}(y_{n}, u_{n}).  (6.4)

It follows that the relation

lim_{n→+∞} \frac{n}{2}∥g(y_{n}, u_{n}) − yn ; K̃∥^{2} = 0,  (6.5)

is valid. From the inequality (4.2) we obtain with ϵ_{n} = \frac{1}{n}

\frac{F (y_{0}, u_{0}) − F (y_{n}, u_{n})}{∥g(y_{n}, u_{n}) − yn ; K̃∥} + \frac{∥( \begin{matrix}y_{n} − y_{0} \\ u_{n} − u_{0} \end{matrix} )∥_{1×2}}{∥q_{n}∥} ≥ −α\frac{< T (y_{n}, u_{n}), q_{n} >_{1×2}}{∥g(y_{n}, u_{n}) − yn ; K̃∥} − \frac{1}{2}∥q_{n}∥
≥^{(6.4)} +\frac{1}{2}∥q_{n}∥.

If the condition (6.3) holds, it follows that the sequence {q_{n}} is bounded. □

Corollary 6.2 Suppose the assumptions of theorem 4.1, (2.6) and (2.7). Then from (4.6) it follows

< T (y_{0}, u_{0}), q > = 0.  (6.6)

Proof : By (4.6)

n · h_{α}(y_{n}, u_{n}) = −α < T (y_{n}, u_{n}), q_{n} > −\frac{1}{2}∥q_{n}∥∥g(y_{n}, u_{n}) − yn ; K̃∥ → 0 as n → +∞

and q_{n} ⇀ q̄ by theorem 6.1. From (6.5) we get with q := αq̄ (6.6), the equation of the state. □ Now define the multifunction B : M ⊂ B_{1} × B_{2} → M again with the abbreviation A := αT (y, u) + Λ(yK̃ − g(y, u)) by

B(y, u) =
\{ \begin{matrix}\{(ỹ, ũ) ∈ M : ⟨A, g′(y, u) (\begin{matrix}ỹ − y \\ ũ − u \end{matrix})⟩ ≤ 0\} & ∀(y, u) ∈ M \\ ∅ & else, \end{matrix}  (6.7)

which possesses the properties

• Dom(B) = M

• (y, u) ∈ B(y, u) ∀(y, u) ∈ M

• B has convex and closed images

Theorem 6.3 Suppose that the assumptions of theorem 6.1 are fulfilled. Then there exists an element q ∈ H such that the following optimality system for the optimal solution (y_{0}, u_{0}) of P

< T (y_{0}, u_{0}), q > = 0  (6.8)
< F′(y_{0}, u_{0}) + [T′(y_{0}, u_{0})]^{∗}q, (\begin{matrix}y − y_{0} \\ u − u_{0} \end{matrix}) >_{1×2} ≥ 0

∀(y, u) ∈ lim sup_{n→+∞)} B(y_{n}, u_{n})

⊂ \{(y, u) ∈ M : ⟨T (y_{0}, u_{0}), g′(y_{0}, u_{0}) ( \begin{matrix}y − y_{0} \\ u − u_{0} \end{matrix} )⟩ = 0\}  (6.9)

Proof : With the multifunction B, (6.7) it follows from the inequality (6.2)

⟨F′(y_{n}, u_{n}), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + nα · ⟨[T′(y_{n}, u_{n})]^{∗}(g(y_{n}, u_{n}) − yn ; K̃), ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2}

+ ⟨( \begin{matrix}Λ_{1}(y_{n} − y_{0}) \\ Λ_{2}(u_{n} − u_{0}) \end{matrix} ) , ( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )⟩_{1×2} + ε_{n} · ∥( \begin{matrix}y − y_{n} \\ u − u_{n} \end{matrix} )∥_{1×2} ≥ 0

∀(y, u) ∈ B(y_{n}, u_{n}). (6.10)

Passing to the limit we get with q := α · q̄ ∈ H the inequality (6.9). By definition the inclusion

lim sup_{n→+∞)} B(y_{n}, u_{n}) ⊂ \{(y, u) ∈ M : ⟨T (y_{0}, u_{0}), g′(y_{0}, u_{0}) ( \begin{matrix}y − y_{0} \\ u − u_{0} \end{matrix} )⟩ ≤ 0\}

is valid. Because of

h_{α}(y_{0}, u_{0}) = 0 ⇒ ⟨T (y_{0}, u_{0}), g(x, v) − g(y_{0}, u_{0})⟩ ≥ 0 ∀(x, v) ∈ M

it follows with (x, v) = (y_{0}, u_{0}) + t(z − y_{0}, w − u_{0}), 0 < t ≤ 1 for all (z, w) ∈ M

⟨T (y_{0}, u_{0}), g′(y_{0}+tτ(z−y_{0}), u_{0}+tτ(w−u_{0})) ( \begin{matrix}z − y_{0} \\ w − u_{0} \end{matrix} )⟩ ≥ 0 ∀(z, w) ∈ M, 0 < τ < 1.

and for t → +0

h_{α}(y_{0}, u_{0}) = 0 ⇒ ⟨T (y_{0}, u_{0}), g′(y_{0}, u_{0}) ( \begin{matrix}y − y_{0} \\ u − u_{0} \end{matrix} )⟩ ≥ 0 ∀(y, u) ∈ M,

also together the inclusion (6.9). □ We check again the optimality system (6.8) and (6.9) by means of example (P2). The first condition (6.8) implies that q = 0, and by (6.9) we get the inequality

−3y + \frac{9}{4} ≥ u ∀(y, u) ∈ lim sup_{n→+∞)} B(y_{n}, u_{n}) ⊂ M

This inequality is valid even for all (y, u) ∈ M = {(y, u) : u ≤ −y^{2}} because spoken geometrically the line u = −3y + \frac{9}{4} is the tangent of the parabola u = −y^{2} at the point (y_{0}, u_{0}) = (\frac{3}{2}, \frac{−9}{4}).

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