Existence Of Optimal Relaxed Control Research Articles

Large Sample Mean-Field Stochastic Optimization

We study a class of sampled stochastic optimization problems, where the underlying state process has diffusive dynamics of the mean-field type. We establish the existence of optimal relaxed controls when the sample set has finite size. The core of our paper is to prove, via $\Gamma$-convergence, that the minimizer of the finite sample relaxed problem converges to that of the limiting optimization problem as the sample size tends to infinity. We connect the limit of the sampled objective functional to the unique solution, in the trajectory sense, of a nonlinear Fokker--Planck--Kolmogorov equation in a random environment. We highlight the connection between the minimizers of our optimization problems and the optimal training weights of a deep residual neural network.

Existence of optimal controls for systems of controlled forward-backward doubly SDEs

AbstractWe wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod{\mathbb{D}}equipped with theS-topology of Jakubowski. Moreover, when the Roxin-type convexity condition is fulfilled, we prove that the optimal relaxed control is in fact strict. Secondly, we prove the existence of a strong optimal controls for a linear forward-backward doubly SDEs. Furthermore, we establish necessary as well as sufficient optimality conditions for a control problem of this kind of systems. This is the first theorem of existence of optimal controls that covers the forward-backward doubly systems.

Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise

Existence theory of optimal relaxed control problem for a class of stochastic hereditary evolution equations driven by Lévy noise has been studied. We formulate the problem in the martingale sense of Stroock and Varadhan to establish existence of optimal controls. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod theorem for nonmetric spaces, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. As application of the abstract theory, Oldroyd and Jeffreys fluids have been studied and existence of optimal relaxed control is established. Existence and uniqueness of a strong solution and uniqueness in law for the two-dimensional Oldroyd and Jeffreys fluids are also shown.

Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics

In this paper we define an infinite-dimensional controlled piecewise deterministic Markov process (PDMP) and we study an optimal control problem with finite time horizon and unbounded cost. This process is a coupling between a continuous time Markov Chain and a set of semilinear parabolic partial differential equations, both processes depending on the control. We apply dynamic programming to the embedded Markov decision process to obtain existence of optimal relaxed controls and we give some sufficient conditions ensuring the existence of an optimal ordinary control. This study, which constitutes an extension of controlled PDMPs to infinite dimension, is motivated by the control that provides Optogenetics on neuron models such as the Hodgkin-Huxley model. We define an infinite-dimensional controlled Hodgkin-Huxley model as an infinite-dimensional controlled piecewise deterministic Markov process and apply the previous results to prove the existence of optimal ordinary controls for a tracking problem.

Existence of optimal controls for systems driven by FBSDEs

We prove the existence of optimal relaxed controls as well as strict optimal controls for systems governed by non linear forward–backward stochastic differential equations (FBSDEs). Our approach is based on weak convergence techniques for the associated FBSDEs in the Jakubowski S-topology and a suitable Skorokhod representation theorem.

Relaxation of nonlinear impulsive controlled systems on Banach spaces

Relaxation control for a class of semilinear impulsive controlled systems is investigated. Existence of mild solutions for semilinear impulsive controlled systems is proved. By introducing a regular countably additive measure, we convexify the original control systems and obtain the corresponding relaxed control systems. The existence of optimal relaxed controls and relaxation results is also proved.

Relaxed Control for a Class of Strongly Nonlinear Impulsive Evolution Equations

Relaxed control for a class of strongly nonlinear impulsive evolution equations is investigated. Existence of solutions of strongly nonlinear impulsive evolution equations is proved and properties of original and relaxed trajectories are discussed. The existence of optimal relaxed control and relaxation results are also presented. For illustration, one example is given.

Relaxed controls for a class of strongly nonlinear delay evolution equations

Relaxed controls for a class of strongly nonlinear delay evolution equations are investigated. Existence of solutions of strongly nonlinear delay equations is proved and properties of original and relaxed trajectories are discussed. The existence of optimal relaxed controls and relaxation result are also presented. For illustration, two examples are given.

Optimal Relaxed Controls for Infinite-Dimensional Stochastic Systems of Zakai Type

In this paper, we present some new results on partially observed control problems for infinite-dimensional stochastic systems in Hilbert space using a fundamental result of Da Prato and Zabczyk on an infinite-dimensional Kolmogorov operator. We prove the existence of optimal relaxed controls for an infinite-dimensional Zakai equation following a semigroup approach and the theory of measurable selections. This result is also extended to differential inclusion. We also present some necessary conditions of optimality.

A class of semilinear stochastic partial differential equations and their controls: Existence results

This paper concerns a class of similinear stochastic partial differential equations, of which the drift term is a second-order differential operator plus a nonlinearity, and the diffusion term is a first-order differential operator. When the nonlinearity is only continuous in the state, it is shown that there exist solutions of the equation provided that the Wiener process involved is one-dimensional. The existence of optimal relaxed controls for this class of equations is also proved. Our method is based on a group analysis of the first-order differential operator and a time change technique.

On the Existence of Optimal Relaxed Controls of Stochastic Partial Differential Equations

This paper is concerned with control problem of systems governed by stochastic partial differential equations, the drift and diffusion terms of which are second- and first-order differential operators, respectively. The existence of an optimal relaxed control is studied in both cases where the systems are degenerate and nondegenerate. It is shown that the higher regularity conditions on the initial state, as required in the existing results, can be dispensed with if the Wiener process is one-dimensional. Some special cases of multidimensional Wiener process are also discussed, which in particular leads to an improvement of a recent result of Bensoussan and Nisio. The method is based on an analysis of the group generated by the first-order differential operator. As an application, an existence theorem of the optimal relaxed control is proved for partially observed diffusions with correlation between the controlled states and the observation noises.