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Introduction
Let K: [0,T] × Rn × Rm → R1, set-valued function F: [0,T] × Rn × Rn → ClConv(Rn) and U ⊂ Rn be given.
Consider the following set-valued optimal control problem (OC): minimize $\int_0^T K(t, x(t), u(t))dt$ , subject to the constraints
(DI) ẋ(t) ∈ F (t, x(t), u(t)), x(t0) = x0, u(t) ∈ U
It is known, that if f is a consistent selection of F and x (·) is a solution to equation
(DE) ẋ(t) = f(t,x(t),u(t)), x(t0) = x0, u(t) ∈ U,
then x (·) is also a solution to inclusion (DI).
Problem
Find at least one selection f of F such that the control problem (OC) subject to the constraints (DE) can be solved.
R. T. Rockafellar considered such type problem in his famous paper [6]. He proved, that convexity of a functional Lf(t, ·, ·) being a minimum of K(t, x, u) over all vectors u ∈ U and such that f(t, x, u) = v is crucial to solve this problem. Therefore, the existence of convex selections of a given multifunction is a key point for solving the problem.
From the Rockafellar’s paper: “...The main justification for our convexity assumptions, however, is that they lead to a theory of duality which would otherwise not be possible...”.
Upper separated set-valued functions
Let X be a Banach space while (Y, ⪯) a Banach lattice, $\overline{Y}=Y\cup\{\pm\infty\}$ . Let F:X → ClConv Y be a set-valued function.
and $\bar{V}(x)=\Pi _{F(x)}(V(x)),\;\bar{W}(x)=\Pi _{F(x)}(W(x))$ ,
where ΠF(x)(a) denotes the projection of a point a onto the set F(x) with ΠF(x)(∞) = ∞.
Definition 1
([2] 2009). A set-valued function F:X → ClConv Y is upper separated if for every x ∈ X and each ε ∈ K+\{0} there exist A ∈ ℒ(X, Y), a ∈ R1 and δ ∈ K+\{0} such that for every $y\in Dom \bar{V}$ and each b ∈ K+ the inequality
This condition means that each point $(x_0,\bar{W}(x_0)-\epsilon )$ can be strictly separated from the set $Epi(\bar{V}):=\{(x,y)\in X\times Y:\bar{V}(x)\leq y\}$ by a hyperplane designed by a continuous linear operator (A, a).
Example 1
Define a set-valued functionF:R → 2R by the formula
F is upper separated, but it is not lower not upper semicontinuous at any point x.
The following results come from [2] and show that the existence of order convex selections of a given multifunction is strictly connected with the "upper separation property".
Theorem 2
LetF:X → ClConv Y. Assume thatFis upper separated with order bounded and order convex values. Then there exists an order convex selectionfofF.
Theorem 3
LetF:X → ClConv Ybe majorized in the neighbourhood of some pointx0. Assume thatFadmits an order convex selectionfsuch that $\bar{W}(x)\preceq f(x)\preceq \bar{V}(x)$ . ThenFis upper separated.
Corollary 4
It is well known that every order convex function majorized in the neighbourhood of at least one pointx0is locally norm Lipschitz. Therefore, the above results are applicable to deduce the existence of solutions of autonomous deterministic and stochastic differential inclusions.
Carathéodory-convex selections of upper separated multifunctions
Let (T, ℳ) be a measurable space satisfying Suslin property, let X be a finite-dimensional Banach space, while Y a separable Banach lattice with an order unit.
Theorem 5
([3]). LetF: T × X → ClConvYbe ${\cal M}\otimes \beta (X)$ -measurable with order-convex and order-bounded values. Assume thatF(t, ·) is upper separated, lower semicontinuous and majorized in a neighbourhood of some pointx1 ∈ Xfor everyt ∈ T. Then there exists a selectionfofFsatisfying
f (·, x) is ℳ -measurable for allx ∈ X,
f(t, ·) is order-convex for everyt ∈ T.
Moreover, the functionf(t, ·) is locally norm Lipschitz for everyt ∈ T.
Note 6
Local compactness ofXas well as separability of the spaceC(X, Y) are crucial for the proof of the above result. Therefore, Theorem 6 can be applied to nonautonomous ordinary differential and stochastic inclusions, but not to retarded functional inclusions. The Fryszkowski’s method used in the proof cannot be applied successfully to the spaceC(C (−r, 0), Rn), which should be considered dealing with retarded inclusions.
Let (𝒮,Σ,μ) be be a σ finite measure space and let X and X* be separable and reflexive Banach spaces. Consider a space (Rd, |·|) with the Euclidean norm. In this space we introduce the canonical order generated by the positive cone K+ = {y ∈ Rd: yi≥0, i = 1, ..., d}. Then (Rd, ⪯) is an order complete Banach lattice with the order unit e = (1, ..., 1). We adjoin to Rd the greatest element +∞ = (+∞, ..., +∞) together with the lowest element −∞ and extend the vector space operations in a natural way. Let $\bar{R^d}=R^d\cup \{\pm \infty\}$ .
Theorem 7
([4]) Let $F:{\cal S}\times X \rightarrow CompConv R^d$ satisfy the following conditions:
F is a Carathéodory set-valued function,
F is upper separated with respect to its second variable.
Then there exists a function f : 𝒮 × X → Rd such that
f(t,x) ∈ F(t,x) for every (t, x),
f (·, ·) is Σ⊗β(X)-measurable,
f(t, ·) is order-convex for every t ∈ 𝒮.
We are able to skip restrictive assumptions on reflexivity of X and X* modyfying the property of upper separability.
Let (𝒮, Σ, μ) be a Σ finite measure space while X a separable Banach space. Let F : 𝒮 × X → ClConvRn be a set-valued function.
Definition 2
([5]). An operator A : 𝒮 × X → Y is linear in the Carathéodory sense if A is Σ-measurable in t, linear in x and such that $\|A(t,x)\|_{R^n} \leq M(t)\|x\|_X$ for some positive and measurable function M. The family of all such nontrivial operators we denote by ℒ(𝒮, X, Y, M).
Definition 3
([5]). F : 𝒮 × X → ClConvRn is upper separated with respect to its last variable if there exist functions $\bar{V}(t,x)\in \Pi _{F(t,x)}(V(t,x))$ , $\bar{W}(t,x)\in \Pi _{F(t,x)}(W(t,x))$ and such that there exists measurable function M(t) such that for every x0 ∈ X there exist A ∈ ℒ(𝒮, X, Rn, M) and a ∈ R1 satisfying for every $y\in X, b\in K^+_{R^n}$ and t ∈ 𝒮 the condition
The existence of solutions of the above Protter’s type stochastic inclusion ensues from Lipschitz continuity of order-convex selections f of F, obtained in Theorem 10. But we know much more on the regularity of such selections by the constructive method of the proof of Theorem 10. They can be taken as convex envelopes (second generalized Fenchel transforms) of a function $\bar{V}_F(t,x)=\Pi _{F(t,x)}(V(t,x))$ . It means that selections f(t, ·) are “greatest” order-convex selections of F(t, ·). Therefore, considering the equation
with a convex envelope f of F, we are able to get additional properties then the ones obtained so far in the differential or stochastic inclusions theory.
For example, taking a sequence (Fn) of set-valued functions satisfying
whereH (·, ·) denotes a Hausdorff distance, there exists a solution X of our stochastic inclusion which is is a uniform on compacts in probability (UCP) limit of solutionsXnof inclusions
Consider again the problem (OR) mentioned at the beginning of the paper i.e.:
Minimize $J(x,u)=\int_0^T K(t, x(t), u(t))dt$ , subject to the constraints
$$
\begin{equation} \dot{x}(t)\in F(t,x(t),u(t)),\;x(t_0)=x_0,\;u(t)\in U
\end{equation}
$$
and find at least one selection f of F such that the optimal control problem subject to the constraints
$$
\begin{equation} \dot{x}(t)= f(t,x(t),u(t)),\;x(t_0)=x_0,\;u(t)\in U
\end{equation}
$$
can be solved.
R. T. Rockafellar considered the problem (OR) in [6]. He defined there a functional Lf(t, x, v) being the minimum of K(t, x, u) over all vectors u ∈ U and such that f(t, x, u) = v. He proved that optimal control problem (OR) is equivalent to minimizing the integral $\int _0^T L_f(t, x(t), \dot{x}(t))dt$ over allx(t0) = x0. The above method is applicable to the reformulated problem (OR) if Lf(t, ·, ·) turn out to be a convex normal integrand in (x, v) for each t ∈ [0, T]. Rockafellar mentioned in his paper that convexity of Lf(t, ·, ·) takes a place in the case of convex K(t, ·, ·), convex U and affine function f(t, ·, ·). Therefore, if we know that our set-valued function F(t, ·, ·) admits at least one affine selection f, then we are able to solve the problem (OR) by a minimization of a convex functional $\int _0^T L_f(t, x(t), \dot{x}(t))dt$ method.
However, the Rockafellar’s method is applicable to the problem (OR) with upper separated multifunctions F as well. Assume that F(t, x, u) is upper separable in (x, u) for every fixed t ∈ [0, T] and Carathéodory in (x, u). Assume that the following growth condition
holds with some positive constant k. Let us take a selection f of F being a convex envelope existing by Theorem 10. Let U be a convex set in Rn and let K(t, ·, ·) be convex and order increasing function with respect to (x, u) for each t ∈ [0, T]. Then we get
Since f is convex and K is convex and order increasing then the function $\tilde{K}(t,x,u))$ is convex with respect to (x, u) for every fixed t ∈ [0, T]. Therefore, our problem can be reformulated equivalently to the following one
and we are in the Rockafellar’s position with convex normal integrand $\tilde{L_f}(t,\cdot ,\cdot)$ , convex U and convex $\tilde{K}(t,\cdot ,\cdot)$ .
The Rockafellar’s method was successfully extended to stochastic optimal control problems by J. M. Bismut in [1].
Let (Ω, ℱ, {ℱt}t∈I, P) be a complete filtered probability space. Assume the filtration (ℱt) satisfies usual hypotheses. Let W = (W(t))t∈I be an Rm-valued Brownian motion defined on this space and adapted to the filtration.
The author considered in [1] the problem of minimizing
