Optimal control problems for certain quasivariational inequalities

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

Abstract. The paper is a first test to derive necessary optimality conditions for problems of optimal control with an abstract quasivariational inequality. The state restriction is represented by a multifunction K : Y ⇝ Y, K(y) = g(y) + K with a convex and closed set K ⊂ Y and a nonlinear operator g : Y → Y. In this case there exists a smooth gap function to describe the quasivariational inequality problem. By means of a penalty method one can derive necessary optimality conditions for this control problem. Key Words : quasivariational inequality, gap function, regularization, Ekeland’s variational principle, penalty method. Mathematics Subject Classification 1991: Primary : 49J52, 49J40 Secondary : 90C26

1 Introduction and statement of the problem

Quasivariational inequality is a generalization of variational inequality in which convex sets involved in the formulation of the variational inequality depend upon the solution of the problem. The problem of solving quasivariational inequality was introduced by Bensoussan and Lions in 1973 (see [5]) in connection with some stochastic impulse control problems, see also [6]. A recent work concerning the quasivariational inequality problem may be found in Mosco [17]. With a quasivariational inequality one can associate in a natural way a family of variational inequalities, see [2, Chapter 11]. The quasivariational inequality problems arise in the applications such as free boundary problems [2], complementarity and equilibrium problems [10], in mathematical physics [18], economics and management [11]. There exists an extensive literatur of many kinds to quasivariational inequality problems. This is concerned with questions about existence and uniqueness of the solution of a quasivariational inequality. Likewise also its numerical solution and the associated problems of optimal control.

Consider the abstract optimal control problem

min \{J(y, v) | \begin{matrix}y ∈ g(y) + K : ⟨T (y, v), z − y⟩_{Y} ≥ 0 ∀z ∈ g(y) + K \\ v ∈ U_{ad} ⊂ U, y ∈ Y \end{matrix} \} (P)

under the following assumptions( summarized by hypothesis (H) ) :

• Y is a Hilbert space, the space of the state constraints, and Y^{∗} its dual; denote by ⟨·, ·⟩_{Y} the duality product between Y and Y^{∗} , by (·, , ·)_{Y} the scalar product of Y and by Λ_{Y} : Y → Y^{∗} the duality mapping.

• U is also a Hilbert space, the space of controls, with the dual space U^{∗} and the duality mapping Λ_{U} : U → U^{∗}. The designation of the scalar- and duality product follows in analogous way. For infinite dimensional problems it is useful the Hilbert spaces Y and U not to identify with its own dual. This is already common and essential for variational inequalities, see [13].

• U_{ad} ⊂ U is a nonempty, convex and closed subset of U, the set of restrictions of the control. Likewise K ⊂ Y is a nonempty, convex and closed subset, and g : Y → Y is a continuously Fréchet-differentiable operator, such that the fixed point set F of the multi-valued operator g(·) + K : Y ⇝ Y is nonempty and convex. The closedness of F is always fulfilled by the assumptions on K and g.

• J : Y × U → ℝ ∪ {+∞} is a proper convex and lower semicontinuous function, which is supposed to be bounded below and Gâteaux-differentiable for all (y, v) ∈ F × U_{ad}.

• T : Y × U → Y^{∗} is an operator, which satisfies:

T is C^{1} in the F réchet sense  (1.1)

and

∀y ∈ F, v ↦ T (y, v) is strongly continuous from U to Y^{∗}.  (1.2)

• Assume that there exists an optimal solution (y^{∗}, v^{∗}) ∈ F × U_{ad} of (P).

For the convenience of the reader we give the

Definition 1.1 ([27, p.555]) Let A : X → X^{∗} be an operator on the real B-space X. A is said to be strongly continuous iff

u_{n} ⇀ u as n → ∞ implies Au_{n} → Au.

The composition K(·) = g(·) + K above is frequently to find by applications, see [2], [17], [19], [20]. At the same time the convex and closed set K is often the cone of the nonnegative elements of an ordered Hilbert space, see the book of Bensoussan & Lions [6]. Here, we find under quasivariational inequalities of elliptic type as an important case the problem with an implicit obstacle of the form

K(φ) := {v | v ≤ M(φ) } ,

where M : V → V is a nonlinear operator in the Hilbert space V. The multifunction K : Y ⇝ Y can be convex or also nonconvex. We have in mind to derive necessary optimality conditions for the abstract control problem (P). It is in general hopeless to expect that a classical mathematical programming approach lead to the usual optimality conditions. Already a problem of optimal control with a variational inequality possesses a degenerate optimality system, see Mignot [15], Mignot & Puel [16], Barbu [3], [4], Bergounioux & Dietrich [7]. Because of the nonconvexity of the problem the regularity condition for a Lagrange multiplier rule is generally not fulfilled. The method to obtain such an optimality system is by Mignot the introduction of the conical derivative and by Barbu the transformation of the variational inequality problem into an operator equation of a multi-valued operator. Shi [23], Wenbin & Rubio [25] and Bergounioux & Dietrich [7] use a penalty method for the optimal control problem and a gap function for grasping the variational inequality problem. We shall transfer the suitable use of the penalty method together with Ekeland’s variational principle and a regularized gap function as in [7] to problems of optimal control with a quasivariational inequality. At the same time we restrict us to multifunctions K : Y ⇝ Y of the form K(y) = g(y) + K, since in this case a regularized gap function is present. If the multifunction K is more general one must firstly find a suitable differentiable and calculable gap function, see [10] because of well-known gap functions for quasivariational inequalities. For the use of several penalty methods and gap functions for bilevel programming problems we refer for example to Marcotte & Zhu [14].

Lemma 1.1 Suppose (H)( already the assumptions F is convex and g ∈ C^{1} are enough) , then

\begin{matrix}y ∈ F, v ∈ U_{ad} : \\ ⟨T (y, v), z − y⟩_{Y} ≥ 0 ∀z ∈ g(y) + K \end{matrix} \} ⟹

\{ \begin{matrix}y ∈ F, v ∈ U_{ad} : \\ ⟨T (y, v), (I − g′(y))(z − y)⟩_{Y} ≥ 0 ∀z ∈ F, \end{matrix}  (1.3)

and

(I − g′(y))(z − y) ∈ T_{K}(y − g(y)) ∀z, y ∈ F,  (1.4)

where T_{K}(·) is the tangent cone of the convex set K.

Proof : From the assumption of (1.3) with z = g(y)+k, k ∈ K and ζ ∈ F, i.e. ζ = g(ζ)+k̄, k̄ ∈ K, it follows that the inequality

⟨T (y, v), ζ − g(ζ) + g(y) − y⟩_{Y} ≥ 0 ∀ζ ∈ F

is true. Further, with ζ = tz + (1 − t)y, z ∈ F and 0 < t < 1, we have by the formula of Lagrange (see [12]) for a number τ ∈ (0, 1)

⟨T (y, v), t(z − y) − g′(y + tτ(z − y))[t(z − y)]⟩_{Y} ≥ 0 ∀z ∈ F, ∀t ∈ (0, 1).

Dividing the last inequality by t > 0 and letting t → +0, we obtain the desired result. Let now y, z ∈ F, then

tz + (1 − t)y ∈ g(tz + (1 − t)y) + K

and further

t(z − y) ∈ g(y + t(z − y)) − g(y) + [K − (y − g(y)]

and as well as by Taylor’s theorem [26, Theorem 4.A]

t(z − y) − t ∫_{0}^{1} g′(y + t · τ · (z − y))(z − y)dτ ∈ K − (y − g(y)).

Dividing by t > 0 and passing to the limit t → +0, we obtain the inclusion (1.4), where T_{K}(·) is the tangent cone of the convex set K. The reverse conclusion to (1.3) is in general not valid, compare example 2.1 below. The variational inequality problem associated to our quasivariational inequality by (1.3) is important in the sequel. The same refers to the inclusion (1.4).

2 The gap function of the quasivariational inequality

In [9], Fukushima has defined a regularized gap function for a variational inequality. We extend his idea to the quasivariational problem

to v ∈ U_{ad} find y ∈ g(y) + K : ⟨T (y, v), z − y⟩_{Y} ≥ 0 ∀z ∈ g(y) + K : (2.1)

For a parameter α > 0, we define the gap function h_{α} : Y × U → ℝ by

h_{α}(y, v) = sup_{z∈g(y)+K} [−α ⟨T (y, v), z − y⟩_{Y} − \frac{1}{2}∥z − y∥_{Y}^{2} ] .  (2.2)

By definition of the metric projection in a Hilbert space, the operator P_{g(y)+K}(·) is representable as projection onto the convex and closed set K by

P_{g(y)+K}(z) = g(y) + P_{K}(z − g(y)),  (2.3)

compare also Noor [19]. The supremum in (2.2) is always uniquely attained by the element

z_{K} = P_{g(y)+K}(y − αΛ_{Y}^{−1}T (y, v)) = g(y) + y_{K}  (2.4)

with

y_{K} = P_{K}(y − g(y) − αΛ_{Y}^{−1}T (y, v)).

The gap function above possesses the following properties.

Theorem 2.1 Suppose (H) and α > 0. Then for the gap function (2.2) hold :

(i) h_{α}(y, v) = \frac{α^{2}}{2} ∥T (y, v)∥_{Y^{∗}}^{2} − \frac{1}{2}∥(I − P_{K})(y − g(y) − αΛ_{Y}^{−1}T (y, v))∥_{Y}^{2} . (ii) h_{α}(y, v) ≥ 0 ∀y ∈ F and v ∈ U_{ad}. (iii) For y ∈ F and v ∈ U_{ad} the equivalences

⟨T (y, v), z − y⟩_{Y} ≥ 0 ∀z ∈ g(y) + K} ⟺ { h_{α}(y, v) = 0,  (2.5)
⟨T (y, v), z − y⟩_{Y} ≥ 0 ∀z ∈ g(y) + K} ⟺ {y = P_{g(y)+K}(y − αΛ_{Y}^{−1}T (y, v)). (2.6)

are valid.

Proof : The properties (i),(ii) follow immediately from the definition of h_{α}. The same holds for the necessity part ′ ⟹′ of the equivalence (2.5) of (iii). We prove now the part (2.5) ′ ⟸′. The equation h_{α}(y, v) = 0 with y ∈ F and v ∈ U_{ad} implies 0 ≥ −α ⟨T (y, v), z − y⟩_{Y} − \frac{1}{2}∥z −y∥_{Y}^{2} for all z ∈ g(y)+K. The set K(y) := g(y)+K ⊂ Y is convex and closed for every fixed y ∈ Y. In the last inequality, we put z = tw+(1−t)y with t ∈ (0, 1] and w ∈ K(y) and get 0 ≥ − α ⟨T (y, v), t(w − y)⟩_{Y} − \frac{t^{2}}{2} ∥w − y∥_{Y}^{2} for all w ∈ K(y). Dividing by t > 0 and t → +0 leads to the fact that (y, v) ∈ F ×U_{ad} is a solution of the quasivariational inequality (2.1). The equivalence (2.6) is well known, see [8]. One can easily see by the hypothesis (H) that the gap function h_{α} is continuously Fréchet-differentiable on Y × U. For the derivative we get

h′_{α}(y, v) = α^{2}[T′(y, v)]^{∗}Λ_{Y}^{−1}T (y, v)

− [( \begin{matrix}I − [g′(y)]^{∗} \\ 0 \end{matrix} ) − α[T′(y, v)]^{∗}Λ_{Y}^{−1}] Λ_{Y} (I − P_{K})(y − g(y) − αΛ_{Y}^{−1}T (y, v)) (2.7)

( see Phelps [22] ). Let now (y^{∗}, v^{∗}) ∈ F × U_{ad} be an optimal solution of (P). Then by (2.4), (2.5) and (2.6) h_{α}(y^{∗}, v^{∗}) = 0, z_{K}^{∗} = P_{g(y^{∗})+K}(y^{∗} − αΛ_{Y}^{−1}T (y^{∗}, v^{∗})) = y^{∗} = g(y^{∗}) − y_{K}^{∗}, hence

y_{K}^{∗} = g(y^{∗}) − y^{∗} and z_{K}^{∗} = y^{∗}  (2.8)

are valid. For the derivative of the gap function we obtain

h′_{α}(y, v) = α^{2}[T′(y, v)]^{∗}Λ_{Y}^{−1}T (y, v)

− [( \begin{matrix}I − [g′(y)]^{∗} \\ 0 \end{matrix} ) − α[T′(y, v)]^{∗}Λ_{Y}^{−1}] Λ_{Y} (y − g(y) − αΛ_{Y}^{−1}T (y, v) − y_{K}). (2.9)

This implies the following connection at the optimal point

h′_{α}(y^{∗}, v^{∗}) = ( \begin{matrix}α(I − [g′(y^{∗})]^{∗})T (y^{∗}, v^{∗}) \\ 0 \end{matrix} ) .  (2.10)

Example 2.1 Consider a simple example for a problem of optimal control of the type (P).

min \{y^{2} + v^{2} | \begin{matrix}y ∈ \frac{3}{8}y^{3} + [−1, +1], v ∈ ℝ : \\ (y + v + 1)(z − y) ≥ 0 ∀z ∈ \frac{3}{8}y^{3} + [−1, +1] \end{matrix} \} . (P1)

For the fixed point set F we get

F = \{y ∈ ℝ |y ∈ \frac{3}{8}y^{3} + [−1, +1]\} = [−2, +2].

We calculate the gap function with α = 1 by theorem 2.1 (i)

h_{1}(y, v) = \frac{1}{2}(y + v + 1)^{2} − \{ \begin{matrix}0 & for & −1 ≤ −\frac{3}{8}y^{3} − v − 1 ≤ +1 \\ \frac{1}{2}(\frac{3}{8}y^{3} + v + 2)^{2} & for & 1 < −\frac{3}{8}y^{3} − v − 1 \\ \frac{1}{2}(\frac{3}{8}y^{3} + v)^{2} & for & −\frac{3}{8}y^{3} − v − 1 < −1 \end{matrix} .

As solutions of the quasivariational inequality problem

y ∈ [−2, +2], v ∈ ℝ : (y + v + 1)(z − y) ≥ 0 ∀z ∈ \frac{3}{8}y^{3} + [−1, +1]

we get the set of points

y = −v − 1 ∀v ∈ [−3, 1] and y = 2 ∀v > −3 as well as y = −2 ∀v < 1,

which are feasible elements for the control problem (P1) above. In the v-y-plane the feasible elements are distributed in the following way

[figure: v; y; 1; 1; -3]

We obtain the uniquely determined optimal solution of (P1) from the picture

y = −v − 1 = v, consequently (y, v) = (−\frac{1}{2}, −\frac{1}{2}),

and min(P) = J(−\frac{1}{2}, −\frac{1}{2}) = \frac{1}{2}.

Consider as a further example the optimal control problem (P2) with the cost function Ĵ(y, v) = (y − 1)^{2} + (v − 1)^{2} and with the same quasivariational inequality problem as (P1). Then obviously min(P2) = Ĵ(2, 1) = 1. The optimal state y^{∗} = 2 is in this case not an interior point of the fixed point set F. Return to the conclusion (1.3) for our example. The associated variational inequality problem is

y ∈ [−2, +2], v ∈ ℝ : (y + v + 1)(1 − \frac{9}{8}y^{2})(z − y) ≥ 0 ∀z ∈ [−2, +2]

and it possesses additionally the solutions (y, v) = (±\frac{2\sqrt{2}}{3}, v), v ∈ ℝ. Therefore (1.3) cannot be an equivalence in general.

3 The penalty method

The properties of the gap function h_{α} imply that the problem (P) is equivalent to

min \{J(y, v) | \begin{matrix}y ∈ g(y) + K : h_{α}(y, v) = 0 \\ v ∈ U_{ad} ⊂ U, y ∈ Y \end{matrix} \} , (P_{α})

where α > 0 is an arbitrary parameter . The problem (P_{α}) is an optimization problem with a smooth equality constraint. It is not possible to apply the usual Lagrange method on this problem since the regularity condition, see for example Zowe and Kurcyusz [28], are even in general not fulfilled. This is connected with the fact that the gap function h_{α} possesses a global minimum even at the points of the solutions of the quasivariational inequality problem. On the other hand h_{α} possesses the properties of a penalty term for the quasivariational inequality problem, so that one can apply the penalty method. Consider therefore the following penalized problem (P_{α}^{n}) of (P_{α})

inf {J_{n}(y, v) |(y, v) ∈ F × U_{ad} } (P_{α}^{n})

with

J_{n}(y, v) := J(y, v) + n · h_{α}(y, v) + \frac{1}{2}∥y − y^{∗}∥_{Y}^{2} + \frac{1}{2}∥v − v^{∗}∥_{U}^{2},

where (y^{∗}, v^{∗}) ∈ F × U_{ad} is an optimal solution of (P). By Ekeland’s variational principle there exists for each ϵ_{n}, n = 1, 2, ... with ϵ_{n} → +0 an element (y_{n}, v_{n}) ∈ F × U_{ad} such that

J(y_{n}, v_{n}) + n · h_{α}(y_{n}, v_{n}) + \frac{1}{2}∥y_{n} − y^{∗}∥_{Y}^{2} + \frac{1}{2}∥v_{n} − v^{∗}∥_{U}^{2} ≤ inf(P_{α}^{n}) + ϵ_{n}

and

J(y_{n}, v_{n}) + n · h_{α}(y_{n}, v_{n}) + \frac{1}{2}∥y_{n} − y^{∗}∥_{Y}^{2} + \frac{1}{2}∥v_{n} − v^{∗}∥_{U}^{2} − ϵ_{n} ∥( \begin{matrix}y − y_{n} \\ v − v_{n} \end{matrix} )∥_{Y ×U} ≤

J(y, v) + n · h_{α}(y, v) + \frac{1}{2}∥y − y^{∗}∥_{Y}^{2} + \frac{1}{2}∥v − v^{∗}∥_{U}^{2} ∀(y, v) ∈ F × U_{ad}.  (3.1)

Theorem 3.1 Suppose (H) and one of the following conditions :

a) The mapping y ↦ T (y, v) is strongly monotone on F uniformly with respect to v ∈ U_{ad}, i.e.

∃m > 0 : ⟨T (y, v) − T (z, v), y − z⟩_{Y} ≥ \frac{m}{2}∥y − z∥_{Y}^{2} ∀y, z ∈ F and ∀v ∈ U_{ad}. (3.2)

It holds α > m^{−1}, T : Y × U → Y^{∗} is a bounded and g : Y → Y a strongly continuous operator.

b) The operator T is weakly-strongly continuous with respect to the both variables y and v, i.e.

y_{n} ⇀ y and v_{n} ⇀ v ⟹ T (y_{n}, v_{n}) → T (y, v)  (3.3)

and the operator g : Y → Y is weakly sequentially continuous, i.e.

z_{n} ⇀ z ⟹ g(z_{n}) ⇀ g(z),

see [27, p.587]. Then, the sequence (y_{n}, v_{n}) generated by Ekeland’s variational principle strongly converges to (y^{∗}, v^{∗}) ∈ F × U_{ad} and moreover

lim_{n→+∞} n · h_{α}(y_{n}, v_{n}) = 0.  (3.4)

Proof : Put (y, v) = (y^{∗}, v^{∗}) ∈ F × U_{ad} into (3.1). It follows

0 ≤ n · h_{α}(y_{n}, v_{n}) + \frac{1}{2}∥y_{n} − y^{∗}∥_{Y}^{2} + \frac{1}{2}∥v_{n} − v^{∗}∥_{U}^{2} ≤

J(y^{∗}, v^{∗}) − J(y_{n}, v_{n}) + ϵ_{n} ∥( \begin{matrix}y_{n} − y^{∗} \\ v_{n} − v^{∗} \end{matrix} )∥_{Y ×U} .  (3.5)

The cost function J : Y × U → ℝ ∪ {+∞} is convex, lower semi-continuous and Gâteaux-differentiable at the optimal point (y^{∗}, v^{∗}) ∈ F × U_{ad}, therefore

0 ≤ n · h_{α}(y_{n}, v_{n}) + \frac{1}{2} ∥( \begin{matrix}y_{n} − y^{∗} \\ v_{n} − v^{∗} \end{matrix} )∥_{Y ×U}^{2} ≤

∥J′(y^{∗}, v^{∗})∥ ∥( \begin{matrix}y_{n} − y^{∗} \\ v_{n} − v^{∗} \end{matrix} )∥_{Y ×U} + ϵ_{n} ∥( \begin{matrix}y_{n} − y^{∗} \\ v_{n} − v^{∗} \end{matrix} )∥_{Y ×U} .  (3.6)

It follows that the sequences {y_{n}} and {v_{n}} are bounded. There exist a weakly convergent subsequences of {y_{n}} and {v_{n}}, again still denoted {y_{n}} and {v_{n}}, such that

y_{n} ⇀ ȳ ∈ F and v_{n} ⇀ v̄ ∈ U_{ad},  (3.7)

because of the convexity and closedness of F and U_{ad}. From (3.6) we see additionally that

h_{α}(y_{n}, v_{n}) → 0 as n → +∞.  (3.8)

a) Now we shall prove that the limit point (ȳ, v̄) ∈ F × U_{ad} is feasible for (P). For the weakly convergent sequences y_{n} ⇀ ȳ and v_{n} ⇀ v̄, we get for any v by the strong monotonicity of T with respect to y

0 ≤ \frac{m}{2}∥y_{n} − ȳ∥_{Y}^{2} ≤ ⟨T (y_{n}, v_{n}) − T (ȳ, v_{n}), y_{n} − ȳ⟩_{Y}

0 ≤ \frac{αm}{2}∥y_{n} − ȳ∥_{Y}^{2} ≤ − α ⟨T (y_{n}, v_{n}), ȳ − y_{n}⟩_{Y} − α ⟨T (ȳ, v_{n}), y_{n} − ȳ⟩_{Y} .

Substituting k̄ = ȳ − g(ȳ) ∈ K we get

\frac{αm − 1}{2}∥y_{n} − ȳ∥_{Y}^{2} ≤ − α ⟨T (y_{n}, v_{n}), k̄ − [y_{n} − g(ȳ)]⟩_{Y} − \frac{1}{2}∥k̄ − [y_{n} − g(ȳ)]∥_{Y}^{2}
−α ⟨T (ȳ, v_{n}), y_{n} − ȳ⟩ ,

and furthermore

\frac{αm − 1}{2}∥y_{n} − ȳ∥_{Y}^{2} ≤ h_{α}(y_{n}, v_{n}) − α ⟨T (ȳ, v_{n}), y_{n} − ȳ⟩_{Y} − \frac{1}{2}∥g(ȳ) − g(y_{n})∥_{Y}^{2}

−(ȳ − g(ȳ) + g(y_{n}) − y_{n}, g(ȳ) − g(y_{n}))_{Y} − α ⟨T (y_{n}, v_{n}), g(ȳ) − g(y_{n})⟩_{Y} .

Since α > m^{−1}, the mapping v → T (y, v) is strongly continuous by (1.2), h_{α}(y_{n}, v_{n}) → 0 by (3.8), the operator g : Y → Y is strongly continuous and T a bounded operator ( [27, p.596]), we obtain the strong convergence of {y_{n}} toward ȳ. From y_{n} → ȳ,v_{n} ⇀ v̄ and the continuity of the projection operator it follows that

y_{K}^{n} := P_{K}(y_{n} − g(y_{n}) − αΛ_{Y}^{−1}T (y_{n}, v_{n})) → ȳ_{K} := P_{K}(ȳ − g(ȳ) − αΛ_{Y}^{−1}T (ȳ, v̄)).

Furthermore by (3.8) and theorem 2.1(i)

0 = lim_{n→+∞} h_{α}(y_{n}, v_{n}) = h_{α}(ȳ, v̄).

Therefore (ȳ, v̄) ∈ F × U_{ad} is a feasible element of (P).

b) In the second case, it follows from (2.2) the representation

h_{α}(y_{n}, v_{n}) = sup_{w∈K}[−α ⟨T (y_{n}, v_{n}), w − (y_{n} − g(y_{n}))⟩_{Y} − \frac{1}{2}∥w − (y_{n} − g(y_{n}))∥_{Y}^{2} ],

where the supremum is attained by the unique element y_{K}^{n} := P_{K}(y_{n} − g(y_{n}) − αΛ_{Y}^{−1}T (y_{n}, v_{n})). By the definition of the metric projection we obtain that

(y_{n} − g(y_{n}) − αΛ_{Y}^{−1}T (y_{n}, v_{n}) − y_{K}^{n}, y − y_{K}^{n})_{Y} ≤ 0 ∀y ∈ K.  (3.9)

Therefore it holds for y = y_{n} − g(y_{n}) ∈ K

∥y_{n} − g(y_{n}) − y_{K}^{n}∥_{Y}^{2} ≤ −α ⟨T (y_{n}, v_{n}), y_{K}^{n} − (y_{n} − g(y_{n})⟩_{Y}

as well as

\frac{1}{2}∥y_{n} − g(y_{n}) − y_{K}^{n}∥_{Y}^{2} ≤ −α ⟨T (y_{n}, v_{n}), y_{K}^{n} − (y_{n} − g(y_{n})⟩_{Y} − \frac{1}{2}∥y_{n} − g(y_{n}) − y_{K}^{n}∥_{Y}^{2}
≤ h_{α}(y_{n}, v_{n}).

Because of (3.8), y_{n} ⇀ ȳ and v_{n} ⇀ v̄, the last inequality implies the strong convergence of y_{n} − g(y_{n}) − y_{K}^{n} → 0. The weak sequential continuity of g implies g(y_{n}) ⇀ g(ȳ) and further y_{K}^{n} ⇀ ȳ − g(ȳ). Consider this in the inequality (3.9). Use further that the operator T is weakly-strongly continuous in both variables. Then we obtain after passing to the limit n → +∞

−α ⟨T (ȳ, v̄), y − (ȳ − g(ȳ))⟩_{Y} ≤ 0 ∀y ∈ K

and therefore h_{α}(ȳ, v̄) ≤ 0. This implies h_{α}(ȳ, v̄) = 0. In both cases, (ȳ, v̄) ∈ F × U_{ad} is a feasible element of (P). Then by (3.5) we get

0 ≤ lim_{n→+∞} [n · h_{α}(y_{n}, v_{n}) + \frac{1}{2}∥y_{n} − y^{∗}∥_{Y}^{2} + \frac{1}{2}∥v_{n} − v^{∗}∥_{U}^{2}] ≤ J(y^{∗}, v^{∗}) − J(ȳ, v̄) ≤ 0.

We also obtain the desired result

y_{n} → y^{∗} = ȳ and v_{n} → v^{∗} = v̄ and n · h_{α}(y_{n}, v_{n}) → 0.  (3.10)

We now want derive the necessary optimality conditions for the penalized problem (P_{α}^{n}). Then we shall pass to the limit n → +∞. Under certain assumptions we shall obtain necessary optimality conditions for (P). Return to the inequality (3.1) and put y = y_{n} + tr, r ∈ F − y_{n}, and v = v_{n} + ts, s ∈ U_{ad} − v_{n}, with t > 0

J(y_{n} + tr, v_{n} + ts) − J(y_{n}, v_{n}) + n[h_{α}(y_{n} + tr, v_{n} + ts) − h_{α}(y_{n}, v_{n})]

+ϵ_{n}t ∥( \begin{matrix}r \\ s \end{matrix} )∥_{Y ×U} + \frac{t^{2}}{2} ∥( \begin{matrix}r \\ s \end{matrix} )∥_{Y ×U} + t (( \begin{matrix}y_{n} − y^{∗} \\ v_{n} − v^{∗} \end{matrix} ) , ( \begin{matrix}r \\ s \end{matrix} ))_{Y ×U} ≥ 0.

Dividing by t > 0 and passing to the limit t → +0, it follows that

⟨J′(y_{n}, v_{n}), ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )⟩_{Y ×U} + n ⟨h′_{α}(y_{n}, v_{n}), ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )⟩_{Y ×U}

+ϵ_{n} ∥( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )∥_{Y ×U} + (( \begin{matrix}y_{n} − y^{∗} \\ v_{n} − v^{∗} \end{matrix} ) , ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} ))_{Y ×U} ≥ 0  (3.11)

for all (z, u) ∈ F × U_{ad}. (2.7) yields for the derivative of the gap function

⟨h′_{α}(y_{n}, v_{n}), ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )⟩_{Y ×U} = α ⟨[T′(y_{n}, v_{n})]^{∗}(y_{n} − g(y_{n}) − y_{K}^{n}), ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )⟩_{Y ×U}

+ ⟨(I − [g′(y_{n})]^{∗})Λ_{Y} (y_{K}^{n} − y_{n} + g(y_{n}) + αΛ_{Y}^{−1}T (y_{n}, v_{n}))), z − y_{n}⟩_{Y} ,

where y_{K}^{n} := P_{K}(y_{n} − g(y_{n}) − αΛ_{Y}^{−1}T (y_{n}, v_{n})). From ( 3.11 ) we get

⟨J′(y_{n}, v_{n}), ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )⟩_{Y ×U} + nα ⟨[T′(y_{n}, v_{n})]^{∗}(y_{n} − g(y_{n}) − y_{K}^{n}), ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )⟩_{Y ×U}

+n ⟨(I − [g′(y_{n})]^{∗})Λ_{Y} (y_{K}^{n} − y_{n} + g(y_{n}) + αΛ_{Y}^{−1}T (y_{n}, v_{n})), z − y_{n}⟩_{Y}

+ (( \begin{matrix}y_{n} − y^{∗} \\ v_{n} − v^{∗} \end{matrix} ) , ( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} ))_{Y ×U} + ϵ_{n} ∥( \begin{matrix}z − y_{n} \\ u − v_{n} \end{matrix} )∥_{Y ×U} ≥ 0 (3.12)

for all (z, u) ∈ F × U_{ad}. With u = v_{n} ∈ U_{ad} and the abbreviations r_{n} := y_{K}^{n} − y_{n} + g(y_{n}) and p_{n} := n · r_{n} we obtain from (3.12) the inequality

⟨Jy^{′}(y_{n}, v_{n}) − α[Ty^{′}(y_{n}, v_{n})]^{∗}p_{n}, z − y_{n}⟩_{Y} + (y_{n} − y^{∗}, z − y_{n})_{Y}
+n ⟨(I − [g′(y_{n})]^{∗})[Λ_{Y} r_{n} + αT (y_{n}, v_{n})], z − y_{n}⟩_{Y} + ϵ_{n}∥z − y_{n}∥_{Y} ≥ 0  (3.13)

for all z ∈ F. Likewise, we get from (3.12) with z = y_{n} ∈ F

⟨Jv^{′}(y_{n}, v_{n}) − α[Tv^{′}(y_{n}, v_{n})]^{∗}p_{n}, u − v_{n}⟩_{U} + ⟨Λ_{U}(v_{n} − v^{∗}), u − v_{n}⟩_{U} + ϵ_{n}∥u − v_{n}∥_{U} ≥ 0 (3.14)

for all u ∈ U_{ad}. Concentrate at first on a simple class of control problems, where the limit process can easily be performed.

Example 3.1 We consider the following control problem

min \{J(y, v) | \begin{matrix}y ∈ y + K : (T (y, v), z − y)_{s} ≥ 0 ∀z ∈ y + K \\ v ∈ U_{ad} ⊂ ℝ^{r}, y ∈ ℝ^{s} \end{matrix} \} (P3)

under the assumption (H) with Y = ℝ^{s}, U = ℝ^{r} and the condition that K ⊂ ℝ^{s} is an arbitrary convex and closed set with 0 ∈ K. It can readily be seen that the fixed point set F coincides with ℝ^{s}. For the optimality system (3.13) and (3.14) we obtain (with α = 1 and r_{n} = y_{K}^{n}) for all z ∈ ℝ^{s}

(Jy^{′}(y_{n}, v_{n}) − n[Ty^{′}(y_{n}, v_{n})]^{∗}(r_{n}), z − y_{n})_{s} + (y_{n} − y^{∗}, z − y_{n})_{s} + ϵ_{n}∥z − y_{n}∥_{s} ≥ 0 (3.15)

and for all u ∈ U_{ad}

(Jv^{′}(y_{n}, v_{n}) − n[Tv^{′}(y_{n}, v_{n})]^{∗}(r_{n}), u − v_{n})_{r} + (v_{n} − v^{∗}, u − v_{n})_{r} + ϵ_{n}∥u − v_{n}∥_{r} ≥ 0. (3.16)

Let the assumptions of theorem 3.1a) be fulfilled. Then we can prove that the sequence {p_{n}} with p_{n} := n·r_{n} possesses a convergent subsequence, again denoted by {p_{n}}. Put z = y_{K}^{n} + y_{n} ∈ ℝ^{s} in (3.15). One gets in the case a):

∥Jy^{′}(y_{n}, v_{n})∥_{s}∥r_{n}∥_{s} + ∥y_{n} − y^{∗}∥_{s}∥r_{n}∥_{s} + ϵ_{n}∥r_{n}∥_{s} ≥ n(Ty^{′}(y_{n}, v_{n})(r_{n}), r_{n})_{s} ≥ n\frac{m}{2}∥r_{n}∥_{s}^{2}. (3.17)

Consequently, in this case {p_{n}} is a bounded sequence and possesses therefore a convergent subsequence p_{n} → p^{∗}. In the case b) of theorem 3.1 we obtain for the inequality (3.17) the form

∥Jy^{′}(y_{n}, v_{n})∥_{s}∥p_{n}∥_{s} + ∥y_{n} − y^{∗}∥_{s}∥p_{n}∥_{s} + ϵ_{n}∥p_{n}∥_{s} ≥ (Ty^{′}(y_{n}, v_{n})(p_{n}), p_{n})_{s} ≥ \frac{m}{2}∥p_{n}∥_{s}^{2},

provided that we additionally suppose the validity of the coercivity of Ty^{′}: there exist m > 0 and ρ > 0, such that

⟨Ty^{′}(y, v)z, z⟩_{Y} ≥ \frac{m}{2}∥z∥_{Y}^{2} ∀(y, v) ∈ F × U_{ad}, ∀z ∈ Y : ∥z∥_{Y} ≥ ρ.

Then {p_{n}} is also a bounded sequence. Now one can carry out the limit n → +∞ of the inequalities (3.15) and (3.16). But at first, we conclude from theorem 3.1

nh_{1}(y_{n}, v_{n}) = −n(T (y_{n}, v_{n}), r_{n})_{s} − \frac{n}{2}∥r_{n}∥_{s}^{2} → +0 as n → +∞.

Consequently, we obtain for the adjoint state p^{∗} the equality (T (y^{∗}, v^{∗}), p^{∗})_{s} = 0. As necessary optimality condition for example 3.1 we obtain under the above conditions: If (y^{∗}, v^{∗}) ∈ ℝ^{s} × U_{ad} is the optimal solution of (P3), then there exists an element p^{∗} ∈ ℝ^{s} such that the following optimality system

(T (y^{∗}, v^{∗}), p^{∗})_{s} = 0
Jy^{′}(y^{∗}, v^{∗}) = [Ty^{′}(y^{∗}, v^{∗})]^{∗}(p^{∗})
(Jv^{′}(y^{∗}, v^{∗}) − [Tv^{′}(y^{∗}, v^{∗})]^{∗}(p^{∗}) , u − v^{∗})_{r} ≥ 0 ∀u ∈ U_{ad} ⊂ ℝ^{r}

holds.

4 Passage to the limit

Always we obtain as a consequence of the optimality system (3.13) and (3.14) the following condition, which is an expression of the fact that the optimal solution (y^{∗}, v^{∗}) is feasible for (P), see lemma 1.1.

Remark 4.1 Suppose (H), α > 0 and the assumptions of theorem 3.1. If (y^{∗}, v^{∗}) ∈ F × U_{ad} is the optimal solution of (P), then there exists a constant K > 0 such that for any n ∈ ℕ

β_{n} := ⟨T (y_{n}, v_{n}), (I − g′(y_{n}))(z − y_{n})⟩_{Y} ≥ − K ∀z ∈ F  (4.1)

holds. Moreover β_{n} → β for n → +∞ with

β := ⟨T (y^{∗}, v^{∗}), (I − g′(y^{∗}))(z − y^{∗})⟩_{Y} ≥ 0 ∀z ∈ F.  (4.2)

Proof : Dividing (3.13) by n > 0, it follows (4.1). Passing to the limit n → +∞, we get the condition (4.2) by theorem 3.1.

Define the following multifunction A : Y × U → 2^{Y} by

A(y, v) = \{ \begin{matrix}\{z ∈ F |⟨αT (y, v) + Λ_{Y} (y_{K} + g(y) − y), (I − g′(y))(z − y)⟩_{Y} ≤ 0\} \\ for all (y, v) ∈ F × U_{ad} \\ ∅ else \end{matrix} . (4.3)

Always holds y ∈ A(y, v) for (y, v) ∈ F × U_{ad}, such that for the domain of A , D(A) = F × U_{ad} is valid. Moreover, the image sets A(y, v) of the multifunction A are convex and closed in Y. If (y^{∗}, v^{∗}) ∈ F × U_{ad} is an optimal solution of (P), then we obtain

A(y^{∗}, v^{∗}) = {z ∈ F | ⟨ T (y^{∗}, v^{∗}), (I − g′(y^{∗}))(z − y^{∗})⟩_{Y} ≤ 0} ,

since y_{K}^{∗} = y^{∗} − g(y^{∗}). Because of (4.2)

A(y^{∗}, v^{∗}) = {z ∈ F | ⟨ T (y^{∗}, v^{∗}), (I − g′(y^{∗}))(z − y^{∗})⟩_{Y} = 0}  (4.4)

holds. Then one may write the inequality (3.13) in the following way

⟨Jy^{′}(y_{n}, v_{n}) − α[Ty^{′}(y_{n}, v_{n})]^{∗}p_{n}, z − y_{n}⟩_{Y} + (y_{n} − y^{∗}, z − y_{n})_{Y} + ϵ_{n}∥z − y_{n}∥_{Y} ≥ 0 (4.5)

for all z ∈ A(y_{n}, v_{n}). If we can prove that {p_{n}} is a bounded sequence, then there exists a subsequence (again denoted by {p_{n}} ), which is weakly convergent, p_{n} ⇀ p^{∗}. Then the element p^{∗} is a socalled adjoint state of the problem, a sort of Lagrange multiplier. Examples shown that one can expect that many problems fulfil this condition. Now we begin by examining the boundedness of the sequence {p_{n}} and consider (3.13). The elements y_{K}^{n} + g(y_{n}) are not necessary located in the set F, although for n → +∞, y_{K}^{n} + g(y_{n}) → y^{∗} ∈ F is fulfilled. Denote by

B_{X}(x^{∗}, ϵ) := {x ∈ X | ∥x − x^{∗}∥ < ϵ}

the open ball with the radius ϵ > 0 and the centre x^{∗} ∈ X in a Hilbert space X. Then we make it a condition of compatibility that there exists a ϵ > 0 such that

y_{K} + g(y) ∈ F  ∀(y, v) ∈ [F ∩ B_{Y} (y^{∗}, ϵ)] × [U_{ad} ∩ B_{U}(v^{∗}, ϵ)].  (4.6)

The condition (4.6) is fulfilled for example 3.1 and also for example 2.1. If y^{∗} is an interior point of F, i.e. y^{∗} ∈ int(F) , then (4.6) is also valid, such that this compatibility condition is often true. For y ∈ F and y_{K} + g(y) ∈ F it follows that with 0 ≤ t ≤ 1

t(y_{K} + g(y)) + (1 − t)y ∈ F

is valid and therefore

y_{t} := t(y_{K} + g(y)) + (1 − t)y − g(t(y_{K} + g(y)) + (1 − t)y) ∈ K.

We denote for fixed (y, v) the following convex hull

K̃(y, v) = conv { y_{t} ∈ K | 0 ≤ t ≤ 1 } .

Theorem 4.1 Suppose (H), the compatibility condition (4.6). Assume further that there exists a ϵ > 0 such that

T (y, v) ∈ N_{K̃(y,v)}(y − g(y)) ∀(y, v) ∈ [F ∩ B_{Y} (y^{∗}, ϵ)] × [U_{ad} ∩ B_{U}(v^{∗}, ϵ)] (4.7)

is valid ( here N_{K̃(y,v)}(·) is the normal cone of the set K̃(y, v) ). For the operator g there exists a k > 0 with

(g′(y)(z), z)_{Y} ≥ − \frac{k}{2}∥z∥_{Y}^{2} ∀z ∈ Y, ∀y ∈ F,  (4.8)

The operators T and g satisfies the conditions of theorem 3.1 for α > \frac{k+2}{m} case a), and in the case b) additional the following coercivity condition for Ty^{′}: there exist m > 0 and ρ > 0 such that

⟨Ty^{′}(y, v)z, z⟩_{Y} ≥ \frac{m}{2}∥z∥_{Y}^{2} ∀(y, v) ∈ F × U_{ad}, ∀z ∈ Y : ∥z∥_{Y} ≥ ρ.  (4.9)

Then there exists an adjoint state p^{∗} ∈ Y such that the following optimality system for (P) is valid:

⟨T (y^{∗}, v^{∗}), p^{∗}⟩_{Y} = 0  (4.10)
⟨Jy^{′}(y^{∗}, v^{∗}) − [Ty^{′}(y^{∗}, v^{∗})]^{∗}(p^{∗}), z − y^{∗}⟩_{Y} ≥ 0 ∀z ∈ lim inf_{(y,v)→(y^{∗},v^{∗})} A(y, v) (4.11)

⟨Jv^{′}(y^{∗}, v^{∗}) − [Tv^{′}(y^{∗}, v^{∗})]^{∗}(p^{∗}) , u − v^{∗}⟩_{U} ≥ 0 ∀u ∈ U_{ad}.  (4.12)

Proof : Let y_{t}^{n} := t(y_{K}^{n} + g(y_{n})) + (1 − t)y_{n} − g(t(y_{K}^{n} + g(y_{n})) + (1 − t)y_{n}) ∈ K and introduce

K̃_{n} = conv { y_{t}^{n} ∈ K | 0 ≤ t ≤ 1} .

Consider T (y_{n}, v_{n}) ∈ NK̃_{n}(y_{n} − g(y_{n})) by (4.7). It follows that

⟨T (y_{n}, v_{n}), k̃ − (y_{n} − g(y_{n}))⟩_{Y} ≤ 0 ∀k̃ ∈ K̃_{n}.

For k̃ = t(y_{K}^{n} + g(y_{n})) + (1 − t)y_{n} − g (t(y_{K}^{n} + g(y_{n})) + (1 − t)y_{n}), with 0 < t < 1, we get

⟨T (y_{n}, v_{n}), tr_{n} − g(y_{n} + tr_{n}) + g(y_{n})⟩_{Y} ≤ 0

and with 0 < τ < 1 by the formula of Lagrange [12]

⟨T (y_{n}, v_{n}), tr_{n} − g′(y_{n} + t · τ · r_{n})(t · r_{n})⟩_{Y} ≤ 0.

We obtain further after division by t > 0 and passing to the limit t → +0

⟨T (y_{n}, v_{n}), (I − g′(y_{n}))(r_{n})⟩_{Y} ≤ 0  (4.13)

Considering the last inequality by (3.13), we obtain with z = y_{K}^{n} + g(y_{n}) ∈ F

⟨Jy^{′}(y_{n}, v_{n}), r_{n}⟩_{Y} + (y_{n} − y^{∗}, r_{n})_{Y} + ϵ_{n}∥r_{n}∥_{Y} ≥

αn ⟨[T′_{y}(y_{n}, v_{n})]^{∗}(r_{n}), r_{n}⟩_{Y} − n∥r_{n}∥_{Y}^{2} + (g′(y_{n})r_{n}, r_{n})_{Y} .  (4.14)

We now use the strongly monotonicity of T with respect to y (3.2) together with the condition (4.8). Then we get from (4.14)

∥Jy^{′}(y_{n}, v_{n})∥_{Y^{∗}}∥r_{n}∥_{Y} + ∥y_{n} − y^{∗}∥_{Y} ∥r_{n}∥_{Y} + ϵ_{n}∥r_{n}∥_{Y} ≥ n\frac{αm − 2 − k}{2} ∥r_{n}∥_{Y}^{2} .

We see that p_{n} = n · r_{n} is bounded, if merely α > \frac{k+2}{m}. In the case b) of theorem 3.1 we get from (4.14)

∥Jy^{′}(y_{n}, v_{n})∥_{Y^{∗}} + ∥y_{n} − y^{∗}∥_{Y} + ϵ_{n} ≥ \frac{α ⟨Ty^{′}(y_{n}, v_{n})p_{n}, p_{n}⟩_{y}}{∥p_{n}∥_{Y}} − ∥p_{n}∥_{Y} + \frac{(g′(y_{n})p_{n}, p_{n})_{Y}}{∥p_{n}∥_{Y}} .

If we assume that ∥p_{n}∥_{Y} → +∞ for a subsequence, then by (4.9) one obtains

∥Jy^{′}(y_{n}, v_{n})∥_{Y^{∗}} + ∥y_{n} − y^{∗}∥_{Y} + ϵ_{n} ≥ \frac{αm − 2 − k}{2} ∥p_{n}∥_{Y} ,

which is not possible for α > \frac{k+2}{m}. Therefore \{p_{n}\} a bounded sequence. Consequently, in both cases there exists a weakly convergent subsequence, again denoted by {p_{n}}, with the weak limit p^{∗} ∈ Y. The limit process of the optimality system (4.5) and (3.14) generates the inequalities (4.11) and (4.12). From theorem 3.1 we know the relation (3.4), also

−nα ⟨T (y_{n}, v_{n}), r_{n}⟩_{Y} − \frac{n}{2}∥r_{n}∥_{Y}^{2} = n · h_{α}(y_{n}, v_{n}) → + 0

for n → +∞. It follows that the equation (4.10) for the adjoint state p^{∗} is valid.

Corollary 4.1 Suppose the assumptions of theorem 4.1. If additionally the multivalued operator A is lower semi-continuous( l.s.c.) at the optimal point (y^{∗}, v^{∗}) of (P), then there exists a p^{∗} ∈ Y, such that

⟨T (y^{∗}, v^{∗}), p^{∗}⟩_{Y} = 0,  (4.15)

⟨Jy^{′}(y^{∗}, v^{∗}) − [Ty^{′}(y^{∗}, v^{∗})]^{∗}(p^{∗}), z − y^{∗}⟩_{Y} ≥ 0

∀z ∈ F : ⟨T (y^{∗}, v^{∗}), (I − g′(y^{∗}))(z − y^{∗})⟩_{Y} = 0,  (4.16)

⟨Jv^{′}(y^{∗}, v^{∗}) − [Tv^{′}(y^{∗}, v^{∗})]^{∗}(p^{∗}) , u − v^{∗}⟩_{U} ≥ 0 ∀u ∈ U_{ad}.  (4.17)

Proof : A is lower semi-continuous at (y^{∗}, v^{∗}), see Aubin & Frankowska [1], iff

A(y^{∗}, v^{∗}) ⊂ lim inf_{(y,v)→(y^{∗},v^{∗})} A(y, v).

Since the set A(y^{∗}, v^{∗}) ⊂ F ⊂ Y is closed, it follows that

A(y^{∗}, v^{∗}) = lim inf_{(y,v)→(y^{∗},v^{∗})} A(y, v).

Use now (4.4) and obtain the claimed optimality system.

5 The lower semi-continuity of A

If we introduce in favour of a simplification the mapping Φ : Y × U → Y^{∗} by

Φ(y, v) = (I − [g′(y)]^{∗})[αT (y, v) + Λ_{Y} (y_{K} + g(y) − y)],

then the multifunction A is representable as

A(y, v) = {z ∈ F | ⟨Φ(y, v), z − y⟩_{Y} ≤ 0}

for all (y, v) ∈ F × U_{ad}. To prove the lower semi-continuity of A at the point (y^{∗}, v^{∗}), we use the way to conclude as by [7]. First we define two set-valued mappings F, G : Y × U → 2^{Y} by

F (y, v) = F − y and G(y, v) = {z ∈ Y | ⟨Φ(y, v), z⟩_{Y} ≤ 0},

then

A(y, v) = [F (y, v) ∩ G(y, v)] + y ∀(y, v) ∈ F × U_{ad}.

Lemma 5.1 Suppose (H), then F and G have convex and closed values and F is even a convex multifunction. Moreover F is lower semi-continuous at any (y, v) ∈ F × U_{ad} and G is lower semi-continuous at any points (y, v) ∈ F × U_{ad} such that Φ(y, v) ≠ 0.

Proof : We prove merely the lower semi-continuity of F and G, the other assertions are obvious. Let be (y, v) ∈ F × U_{ad}, consider a sequence (y_{n}, v_{n}) ∈ F × U_{ad} converging to (y, v) and choose z ∈ F − y. The sequence

z_{n} := z + y − y_{n} ∈ F − y_{n}

converges to z ∈ F − y . Consequently, F is lower semi-continuous at any point (y, v) ∈ F × U_{ad}. Let be again (y, v) ∈ F × U_{ad} and z ∈ G(y, v). Consider a sequence (y_{n}, v_{n}) ∈ F × U_{ad} converging to (y, v). If ⟨Φ(y, v), z⟩_{Y} < 0, then by the continuity of the mapping Φ at the point (y, v) it follows that

⟨Φ(y_{n}, v_{n}), z⟩_{Y} < 0

for all n large enough. We can choose in this case z_{n} = z. If ⟨Φ(y, v), z⟩_{Y} = 0, then α_{n} := ⟨Φ(y_{n}, v_{n}), z⟩_{Y} → 0 as n → +∞. We set

z_{n} = \{ \begin{matrix}z − |α_{n}| · \frac{Φ(y_{n},v_{n})}{∥Φ(y_{n},v_{n})∥_{Y}^{2}} & for & Φ(y_{n}, v_{n}) ≠ 0 \\ z & for & Φ(y_{n}, v_{n}) = 0, \end{matrix}

then z_{n} → z and

⟨Φ(y_{n}, v_{n}), z_{n}⟩_{Y} = \{ \begin{matrix}α_{n} − |α_{n}| & for & Φ(y_{n}, v_{n}) ≠ 0 \\ 0 & for & Φ(y_{n}, v_{n}) = 0. \end{matrix}

Since always α_{n}−|α_{n}| ≤ 0, it follows that z_{n} ∈ G(y_{n}, v_{n}) and G is lower semi-continuous at the points (y, v) ∈ F × U_{ad} with Φ(y, v) ≠ 0. Here is a result of Penot 1993 [21] on the persistence of the lower semi-continuity of multifunctions under intersection.

Proposition 5.1 Let F , G : Y × U → 2^{Y} be two convex-valued multifunctions which are lower semi-continuous at (y^{∗}, v^{∗}) , assume that F is a convex multifunction with a closed image and that int(F (y^{∗}, v^{∗})) is nonempty. Then

G(y^{∗}, v^{∗}) ∩ int(F (y^{∗}, v^{∗})) ≠ ∅ ⟹ F ∩ G is l.s.c. at (y^{∗}, v^{∗}).

Now we may formulate the following

Theorem 5.1 Suppose the assumptions of theorem 4.1, that F has a nonempty interior and (y^{∗}, v^{∗}) be an optimal solution of (P) . Then, there exists p^{∗} ∈ Y such that (4.10) and (4.12) are fulfilled, where either (I − [g′(y^{∗})]^{∗})T (y^{∗}, v^{∗}) = 0, or

∀z ∈ F : ⟨(I − [g′(y^{∗})]^{∗})T (y^{∗}, v^{∗}), z − y^{∗}⟩_{Y} = 0 ⟹
⟨Jy^{′}(y^{∗}, v^{∗}) − [Ty^{′}(y^{∗}, v^{∗})]^{∗}(p^{∗}), z − y^{∗}⟩_{Y} ≥ 0.  (5.1)

Proof : Since Φ(y^{∗}, v^{∗}) = (I − [g′(y^{∗})]^{∗})T (y^{∗}, v^{∗}), it follows from Φ(y^{∗}, v^{∗}) ≠ 0 that we can apply proposition 5.1 and that the multifunction A is l.s.c. at (y^{∗}, v^{∗}). By theorem 4.1 we obtain the optimality system (4.10), (4.12) and instead of (4.11) the condition (5.1).

6 Applications

An application to problems of optimal control in finite dimensional spaces is easily possible. Here the required assumptions are immediately to verify. For example, on this way one can write down the necessary conditions for the problems (P1) and (P2) as well as for other finite dimensional problems of optimal control and confirm the optimality conditions. In the infinite dimensional case, applications are slightly more difficult. However, regard for example a control problem, where T (y, v) = Ay + Bv + f, A is a linear uniformly elliptic differential operator A and B is a linear and compact operator B fulfilling the assumptions. The necessity to demand the strong continuity of the operator g : Y → Y is here a restrictive condition. One can weaken the condition to the strong continuity of g at the points of a set F ∩ B_{Y} (y^{∗}, ϵ) for an ϵ > 0. Consider a infinite dimensional example of the described type, where one may confirm the optimality conditions.

Example 6.1

We assume

• Y = H_{0}^{1}(0, 1), K := {y(x) |y(x) = λsin(πx) , λ ∈ [0, 1]} = [0, 1]sin(πx) ⊂ Y is a closed and convex subset,

• j : H_{0}^{1}(0, 1) → L^{2}(0, 1) is the embedding operator having the adjoint embedding operator j^{∗} : L^{2}(0, 1) → H^{−1}(0, 1),

• U = L^{2}(0, 1), U_{ad} := {v(x) ∈ L^{2}(0, 1) | v(x) = c ∈ ℝ} ⊂ U is the subset of the feasible controls,

• g : H_{0}^{1}(0, 1) → H_{0}^{1}(0, 1) is the following strongly continuous operator

g(y) = (sin(πx), y(x))_{H_{0}^{1}(0,1)} · sin(πx).

Then, for the set of fixed points F of the set-valued operator K : Y ⇝ Y with K(y) = g(y) + K we get

F = [\frac{−2}{π^{2} − 2}, 0] · sin(πx).

Consider the following controlled quasivariational inequality problem :

F ind y(x) ∈ F ⊂ H_{0}^{1}(0, 1) and v(x) ∈ U_{ad} ⊂ L^{2}(0, 1), such that

⟨−y′′(x) + j^{∗}v(x), z(x) − y(x)⟩_{H_{0}^{1}(0,1)} ≥ 0 ∀z(x) = [π ∫_{0}^{1} cos(πx)y′(x)dx + k] sin(πx) (6.1)

with k ∈ [0, 1]. As cost function we choose the strongly convex and Fréchet-differentiable function

J(y, v) = \frac{1}{2}∥y(x) − x(x − 1)∥_{H_{0}^{1}(0,1)}^{2} + \frac{1}{2}∥v(x) − 2∥_{L^{2}(0,1)}^{2}.  (6.2)

If we set (y(x), v(x)) = (αsin(πx), c) ∈ F × U_{ad} with α ∈ [\frac{−2}{π^{2}−2}, 0] and c ∈ ℝ, then we get for the quasivariational inequality problem

(\frac{α}{2}π^{2} + k − α) (c + \frac{π^{3}α}{4}) ≥ 0 ∀k ∈ [0, 1].

By means of the (nonconvex) set of solutions of the quasivariational inequality problem we obtain

( y^{∗}(x), v^{∗}(x) ) = ( \frac{−2}{π^{2} − 2}sin(πx) , \frac{π^{3}}{2π^{2} − 4} ),

as the uniquely determined optimal solution of the control problem. The minimal value of J is J(y^{∗}, v^{∗}) = 0.028978... . Now we check the necessary optimality conditions. From the optimality condition to the control we obtain

∫_{0}^{1} p^{∗}(x)dx = \frac{π^{3}}{2π^{2} − 4} − 2.  (6.3)

The adjoint equation reads

∫_{0}^{1} (\frac{−2π^{2}}{π^{2} − 2}sin(πx) + \frac{π^{3}}{2π^{2} − 4}) p^{∗}(x)dx = 0,

therefore

∫_{0}^{1} p^{∗}(x)dx = \frac{4}{π} ∫_{0}^{1} p^{∗}(x)sin(πx)dx.

(6.3) yields

∫_{0}^{1} p^{∗}(x)sin(πx)dx = \frac{π}{4} (\frac{π^{3}}{2π^{2} − 4} − 2) = −0.02356... .  (6.4)

Furthermore, for z(x) = αsin(πx) ∈ F it follows that

(I − g′(y^{∗}))(z − y^{∗}) = (α + \frac{2}{π^{2} − 2})(1 − \frac{π^{2}}{2} )sin(πx)

and because of that

{z ∈ F | ⟨T (y^{∗}, v^{∗}), (I − g′(y^{∗}))(z − y^{∗})⟩_{H_{0}^{1}(0,1)} = 0} = F.

As optimality condition of the state it follows

∫_{0}^{1} (\frac{−2π^{2}}{π^{2} − 2}sin(πx) + 2 + (p^{∗})′′(x)) (α + \frac{2}{π^{2} − 2})sin(πx)dx ≥ 0

∀α ∈ [\frac{−2}{π^{2}−2}, 0] and further that

∫_{0}^{1} p^{∗}(x)sin(πx)dx ≤ \frac{−1}{π^{2} − 2} + \frac{4}{π^{3}} = 0.00193494... ,

in accordance with (6.4). Therefore there exists an adjoint state p^{∗} ∈ H_{0}^{1}(0, 1) with the required optimality conditions.

Acknowledgement. The author would like to thank the referees for their careful review of the manuscript and for the helpful comments and suggestions which they provided.

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