Forward-backward evolution equations and applications

Abstract

Well-posedness is studied for a special system of two-point boundary value problem for evolution equations which is called a forward-backward evolution equation (FBEE, for short). Two approaches are introduced: A decoupling method with some brief discussions, and a method of continuation with some substantial discussions. For the latter, we have introduced Lyapunov operators for FBEEs, whose existence leads to some uniform a priori estimates for the mild solutions of FBEEs, which will be sufficient for the wellposedness. For some special cases, Lyapunov operators are constructed. Also, from some given Lyapunov operators, the corresponding solvable FBEEs are identified. ). 1 considerations. Whereas, in this paper, we are interested in the well-posedness of (1.1). On the other hand, our system has a special structure, involving one forward evolution equation and one backward evolution equation. Hence, we use the name FBEE to distinguish the current situation from other situations in the literature.A pair of functions (y(·), ψ(·)) is called a strong solution of (1.1) if these functions are differentiable almost everywhere, with the propertyand the equations are satisfied almost everywhere. A pair (y(·), ψ(·)) is called a mild solution (or a weak solution) to FBEE (1.1) if the following system of integral equations are satisfied:Note that in the case A is bounded, (1.1) and (1.2) are actually equivalent, and thus, a mild solution (y(·), ψ(·)) is actually a strong solution.Our study of the above system is mainly motivated by the study of optimal control theory. It is known that for a standard optimal control problem of an evolution equation with, say, a Bolza type cost functional, by applying the Pontryagin maximum/minimum principle, one will obtain an optimality system of the above form whose solution will give a candidate for the optimal trajectory and its adjoint ([14]). Therefore, solvability of the above type system is important, at least for optimal control theory of evolution equations.Roughly speaking, when T is small enough, or the Lipschitz constants of the involved functions are small enough, one can show that FBEE (1.1) will have a unique mild solution, by means of contraction mapping theorem. On the other hand, if (1.1) is the optimality system (obtained via Pontryagin maximum/minimum principle) of a corresponding optimal control problem which admits an optimal control, then this FBEE admits a mild solution, which might not be unique. Further, if the corresponding optimal control has an optimal control and the optimality system admits a unique mild solution, then this solution can be used to construct the optimal control(s). Hence, under proper conditions, FBEE (1.1) could admit a (unique) mild solution, without restriction on the length of the time horizon T , and/or the size of the Lipschitz constants of the involved functions. This is actually the case if the FBEE is the optimality system of a linear-quadratic (LQ, for sort) optimal control problem satisfying proper conditions ([14]).In this paper, we ...

Solvability of an Operator Riccati Integral Equation in a Reflexive Banach Space

“…In these papers, the solvability of an autonomous Riccati equation in operator functions ranging in the spaces L(X * , X) and L(X, X * ), respectively, was established. In the papers [8,9], the solvability of the Riccati equation was used to prove the solvability of systems of forward-backward evolution equations.…”