1,720,988 research outputs found
Stochastic maximum principle for SPDEs with noise and control on the boundary
Full text linkIn this paper we prove necessary conditions for optimality of a stochastic control problem for a class of
stochastic partial differential equations that is controlled through the boundary. This kind of problem can
be interpreted as a stochastic control problem for an evolution system in a Hilbert space. The regularity
of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays
a crucial role in the formulation of the maximum principle
On the Backward Stochastic Riccati Equation in Infinite Dimensions
No full textThis paper deals with linear quadratic stochastic control problems in infinite-dimensional Hilbert space with random operator valued coefficients. This is certainly a classical problem, first treated by J.-M. Bismut [SIAM J. Control Optimization 14 (1976), no. 3, 419–444; MR0406663 (53 #10449)] for general finite-dimensional problems and later by N. U. Ahmed [SIAM J. Control Optim. 19 (1981), no. 3, 401–430; MR0613102 (82j:93054)] for infinite-dimensional problems involving unbounded random operator valued coefficients. Following a basic variational principle one can easily derive the necessary conditions for optimality. This approach gives rise to backward stochastic differential equations on infinite-dimensional spaces. The questions of existence, even uniqueness, and regularity properties of solutions of the associated operator Riccati equations are of fundamental importance. Such problems have been treated in [N. U. Ahmed, op. cit.] for operators based on the Gel′fand triple {V, H, V ∗}. In the paper under review the operators include a deterministic infinitesimal generator of a C0 semigroup and some bounded linear operator valued stochastic processes. The paper presents some interesting results on the existence and uniqueness of solutions of the stochastic Riccati equations. Conceptually one may consider this as the solution of the LQR problem in infinite-dimensional Hilbert space, though from a practical point of view this is far from it. It seems the authors have missed some earlier papers in the same area which we included here (see the papers cited above as well as [N. U. Ahmed, in Differential equations and applications, Vol. I, II (Columbus, OH, 1988), 13–19, Ohio Univ. Press, Athens, OH, 1989; MR1026110 (91h:60067)])
Stochastic Partial Differential Equations in Bounded domains with Dirichlet Boundary Conditions
No full textOptimal control of two scale stochastic systems in infinite dimensions: the BSDE approach
Full text linkIn this paper we study, by probabilistic techniques, the convergence of the value function for a two-scale, infinite-dimensional, stochastic controlled system as the ratio between the two evolution speeds diverges.
The value function is represented as the solution of a backward stochastic differential equation (BSDE) that it is shown to converge towards a reduced BSDE. The noise is assumed to be additive both in the slow and the fast equations for the state. Some non degeneracy condition on the slow equation are required. The limit BSDE involves the solution of an ergodic BSDE and is itself interpreted as the value function of an auxiliary stochastic control problem on a reduced state space
Ergodic optimal quadratic control for an affine equation with stochastic and stationary coefficients
Full text linkWe study ergodic quadratic optimal stochastic control problems for an affine state equation with state and control dependent noise and with stochastic coefficients. We assume stationarity of the coefficients and a finite cost condition. We first treat the stationary case and we show that the optimal cost corresponding to this ergodic control problem coincides with the one corresponding to a suitable stationary control problem and we provide a full characterization of the ergodic optimal cost and control
Singular Limit of Two-Scale Stochastic Optimal Control Problems in Infinite Dimensions by Vanishing Noise Regularization
Full text linkIn this paper we study the limit of the value function for a two-scale, infinitedimensional, stochastic controlled system with cylindrical noise and possibly degenerate diffusion. The limit is represented as the value function of a new reduced control problem (on a reduced state space). The presence of a cylindrical noise prevents representation of the limit by viscosity solutions of Hamilton-Jacobi-Bellman equations as in [Swiech, ESAIM Control Optim. Calc. Var., to appear] while degeneracy of diffusion coefficients prevents representation as a classical backward stochastic differential equation as in [Guatteri and Tessitore, Appl. Math. Optim., 83 (2021), pp. 1025-1051]. We use a vanishing noise""regularization technique
On the Existence of Optimal Controls for SPDEs with Boundary Noise and Boundary Control.
Full text linkWe consider a stochastic optimal control problem for an heat equation with
boundary noise and boundary control. Under suitable assumptions on the coefficients, we
prove existence of optimal controls in strong sense by solving the stochastic hamiltonian system related
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