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<p>We study an optimal control problem for a&#xa0;stochastic system governed by a&#xa0;second-order stochastic partial differential equation of hyperbolic type with Goursat boundary conditions.The&#xa0;controls are assumed to be measurable and bounded.We consider the case where a&#xa0;two-parameter “white noise” appears on the right-hand side of a&#xa0;controlled system of second-order nonlinear hyperbolic equations.The goal is to minimize the expectation of the quality functional evaluated at the terminal point of the domain.Such problems arise, for instance, in modeling drying, sorption, and other processes subject to random influences represented by the standard two-parameter “white noise” in the plane.Using a&#xa0;modified version of the increment method,we derive a&#xa0;second-order increment formula for the quality functional,which allows us to establish first-order necessary conditions of the linearized Pontryagin-type maximum principle andto analyze quasisingular controls,that is, controls for which the first-order necessary condition degenerates.We formulate necessary conditions of first- and second-order optimality.Finally, employing a&#xa0;special variation of the control,we derive a&#xa0;pointwise necessary condition for the optimality of quasisingular controls.</p>

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The Problem of Quasisingular Optimal Controls in Stochastic Hyperbolic Systems

  • 期刊论文

  • 发表日期：

    2025年7月23日

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  • K. B. Mansimov,
  • R. O. Mastaliev

摘要

We study an optimal control problem for a stochastic system governed by a second-order stochastic partial differential equation of hyperbolic type with Goursat boundary conditions.The controls are assumed to be measurable and bounded.We consider the case where a two-parameter “white noise” appears on the right-hand side of a controlled system of second-order nonlinear hyperbolic equations.The goal is to minimize the expectation of the quality functional evaluated at the terminal point of the domain.Such problems arise, for instance, in modeling drying, sorption, and other processes subject to random influences represented by the standard two-parameter “white noise” in the plane.Using a modified version of the increment method,we derive a second-order increment formula for the quality functional,which allows us to establish first-order necessary conditions of the linearized Pontryagin-type maximum principle andto analyze quasisingular controls,that is, controls for which the first-order necessary condition degenerates.We formulate necessary conditions of first- and second-order optimality.Finally, employing a special variation of the control,we derive a pointwise necessary condition for the optimality of quasisingular controls.