Stochastic Control on Hilbert Space for Linear Evolution Equations with Random Operator-Valued Coefficients

This chapter is devoted to presenting some results on controllability for forward and backward stochastic linear evolution equations with possibly unbounded control operators. By the duality argument, these controllability problems can be reduced to suitable observability for the dual equations. Explicit forms of controls for the controllability problems are also provided. Finally, the controllability of some forward stochastic evolution equations are shown to be equivalent to that of suitably chosen backward stochastic evolution equations.

Let K and H be two separable Hilbert spaces and be a cylindrical Wiener process with values in K defined on a probability space denote its natural filtration. Given , we look for an adapted pair of process with values in H and respectively is defined in §1),which solves a semilinear stochastic evolution equation of the backward form: where A is the infinitesimal generators of a C 0-semigroup {eAt } on H. The precise meaning of the equation is A linearized version of that equation appears in infinite-dimensional stochastic optimal control theory as the equation satisfied by the adjoint process. We also give our results to the following backward stochastic partial differential equation:

We construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certain form of N-agent stochastic differential game (SDG) of a mean-field type. The information structure considered for the SDG is of a possible asymmetric and partial type. To prove our SMP we take an approach based on spike-variations and adjoint representation techniques, analogous to that of S. Peng (SIAM J. Control Optim. 28(4):966–979, 1990) in the optimal stochastic control context. In our proof we apply adjoint representation procedures at three points. The first-order adjoint processes are defined as solutions to certain mean-field backward stochastic differential equations, and second-order adjoint processes of a first type are defined as solutions to certain backward stochastic differential equations. Second-order adjoint processes of a second type are defined as solutions of certain backward stochastic equations of a type that we introduce in this paper, and which we term conditional mean-field backward stochastic differential equations. From the resulting representations, we show that the terms relating to these second-order adjoint processes of the second type are of an order such that they do not appear in our final SMP equations. A comparable situation exists in an article by R. Buckdahn, B. Djehiche, and J. Li (Appl. Math. Optim. 64(2):197–216, 2011) that constructs a SMP for a mean-field type optimal stochastic control problem; however, the approach we take of using these second-order adjoint processes of a second type to deal with the type of terms that we refer to as the second form of quadratic-type terms represents an alternative to a development, to our setting, of the approach used in their article for their analogous type of term.

This paper addresses general discounted stochastic linear-quadratic (LQ) optimal control problems, where the control cost weighting operator may be singular, and the system is governed by controlled stochastic evolution equations in a Hilbert space. In the time-consistent setting, we establish conditions on the problem coefficients that guarantee the well-posedness of the generalized Riccati equation, despite the singularity of the control weighting operator. For the time-inconsistent case, a necessary and sufficient condition for the existence of a closed-loop equilibrium operator is established in terms of the solvability of the equilibrium Riccati evolution equation with certain regularity. In the last part of the paper, the solvability of the equilibrium Riccati evolution equation is established through the application of the multi-person differential game method. These results extend the recent work of Lü and Ma [Science China Mathematics, 67 (2024), 211–236], which addressed this problem for controlled stochastic differential equations.

We consider a stochastic control problem which is composed of a controlled stochastic differential equation and whose associated cost functional is defined through a controlled backward stochastic differential equation. Under appropriate convexity assumptions on the coefficients of the forward and the backward equations, we prove the existence of an optimal control on a suitable reference stochastic system. The proof is based on an approximation of the stochastic control problem by a sequence of control problems with smooth coefficients, admitting an optimal feedback control. The quadruplet formed by this optimal feedback control and the associated solution of the forward and the backward equations are shown to converge in law, at least along a subsequence. The convexity assumptions on the coefficients then allow to construct from this limit an admissible control process which, on an appropriate reference stochastic system, is optimal for our stochastic control problem.

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

Stochastic evolution equations with general white noise disturbance

A stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space is considered. Thanks to the identification of the mild solution of the state equation as a $\nu$-weak Dirichlet process, the value process is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is used to prove a verification theorem. Through that technique several of the required assumptions are milder than those employed in previous contributions about nonregular solutions of Hamilton--Jacobi--Bellman equations.

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

Solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations. Parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control.

A stochastic control problem is formulated and we get the explicit form of the optimal control for initial-data-parameterized linear quadratic stochastic optimal control problems with random jumps. The optimal control can be proved to be unique. A stochastic Riccati equation is rigorous derived from the stochastic Hamilton system, which provides an optimal feedback control. This completes the the interrelationship between the stochastic Riccati equation and stochastic Hamilton system as two different but equivalent tools for the stochastic linear quadratic problem.

This paper investigates the optimal stabilisation control of discrete time mean-field Markov jump system with multiplicative noises in the finite and infinite horizon. The forward and backward stochastic difference equation as well as the newly defined coupled Riccati equation, are used to ascertain the necessary and sufficient conditions for the solvability of the finite horizon problem and then the optimal controller and performance index are derived. For the infinite case, a new Lyapunov function based on the optimal performance index is defined and one coupled algebraic Riccati equation (CARE) is developed, by which the necessary and sufficient stabilisation condition and the infinite optimal solution are derived. The mean-field stochastic Markov jump system is mean square stabilizable if and only if the CARE has a unique positive semi-definite (positive definite) solution. The main techniques employed in this paper are the establishment of the maximum principle and the construction of the Lyapunov function.

We consider a forward-backward system of stochastic evolution equations in a Hilbert space. Under nondegeneracy assumptions on the diffusion coefficient (that may be nonconstant) we prove an analogue of the well-known Bismut-Elworthy formula. Next, we consider a nonlinear version of the Kolmogorov equation, i.e. a deterministic quasilinear equation associated to the system according to Pardoux, E and Peng, S. (1992). "Backward stochastic differential equations and quasilinear parabolic partial differential equations". In: Rozowskii, B.L., Sowers, R.B. (Eds.), Stochastic Partial Differential Equations and Their Applications , Lecture Notes in Control Inf. Sci., Vol. 176, pp. 200-217. Springer: Berlin. The Bismut-Elworthy formula is applied to prove smoothing effect, i.e. to prove existence and uniqueness of a solution which is differentiable with respect to the space variable, even if the initial datum and (some) coefficients of the equation are not. The results are then applied to the Hamilton-Jacobi-Bellman equation of stochastic optimal control. This way we are able to characterize optimal controls by feedback laws for a class of infinite-dimensional control systems, including in particular the stochastic heat equation with state-dependent diffusion coefficient.

A stochastic minimax optimal time-delay state feedback control strategy for uncertain quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal state feedback control problem of uncertain quasi-integrable Hamiltonian system with time delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed state feedback control forces are approximated by the control forces without time delay and the original problem is converted into a stochastic optimal state feedback control problem of an uncertain system without time delay. After that, by following a procedure based on the stochastic averaging method and stochastic differential game, the worst-case disturbances and the optimal controls are obtained from solving a Hamiltonian–Jacobi–Isaacs (HJI) equation. As an example, the stochastic minimax optimal state feedback control of a Duffing oscillator with parametric disturbances and time-delayed feedback control is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.

This paper endeavours to directly establish the observability inequality of backward stochastic heat equations involving a Lévy process for measurable sets. As an immediate application, we attain the approximate controllability of forward stochastic heat equations featuring a Lévy process. Subsequently, we embark upon the formulation of a nuanced optimization problem, encompassing both the determination of an optimal actuator location and its associated minimum norm control. This formulation is elegantly cast as a two-person zero-sum game problem. We then proceed to articulate a set of conditions–both sufficient and necessary–required for optimizing the solution through the prism of a Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.