On the maximum principle for relaxed control problems of nonlinear stochastic systems

• Published in: Advances in continuous and discrete models Vol. 2024; no. 1; p. 8
• Main Authors: Mezerdi, Meriem, Mezerdi, Brahim
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 20.03.2024; Springer Nature B.V
• Subjects: Analysis; Approximation; Brownian motion; Continuity (mathematics); Convexity; Difference and Functional Equations; Differential equations; Diffusion; Diffusion coefficient; Drift; Functional Analysis; Martingales; Mathematics; Mathematics and Statistics; Maximum principle; Nonlinear control; Nonlinear systems; Optimal control; Optimization; Ordinary Differential Equations; Partial Differential Equations; Stochastic systems; Weak convergence; 93E20; 60H10; 60F99; 60H30; Stochastic optimal control; Adjoint process; Relaxed control; Maximum principle; Tightness; Stochastic differential equation; Martingale measure
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-024-03803-w

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng’s maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable.