Minimax and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Time-Delay Systems

The paper deals with a Bolza optimal control problem for a dynamical system which motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional in this problem, the Cauchy problem for the Hamilton-Jacobi-Bellman equation with coinvariant derivatives is considered. Minimax and viscosity solutions of this problem are studied. It is proved that both of these solutions exist, are unique and coincide with the value functional.

[1]  Frank H. Clarke,et al.  Mean value inequalities in Hilbert space , 1994 .

[2]  A. I. Subbotin,et al.  Game-Theoretical Control Problems , 1987 .

[3]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[4]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[5]  Peter R. Wolenski,et al.  Necessary conditions for functional differential inclusions , 1996 .

[6]  A. N. Krasovskii,et al.  Control under Lack of Information , 1994, Dynamics and Control.

[7]  R. Newcomb.  VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS , 2010 .

[8]  G. Prato,et al.  Some results on non-linear optimal control problems and Hamilton-Jacobi equations in infinite dimensions , 1990 .

[9]  Jiongmin Yong,et al.  Optimal Control Theory for Infinite Dimensional Systems , 1994 .

[10]  E. N. Barron,et al.  Application of viscosity solutions of infinite-dimensional Hamilton-Jacobi-Bellman equations to some problems in distributed optimal control , 1990 .

[11]  N. N. Krasovskii,et al.  On the analytic construction of an optimal control in a system with time lags , 1962 .

[12]  Harvey Thomas Banks,et al.  Necessary Conditions for Control Problems with Variable Time Lags , 1968 .

[13]  A. I. Subbotin.  ON A PROPERTY OF THE SUBDIFFERENTIAL , 1993 .

[14]  H. Soner.  On the Hamilton-Jacobi-Bellman equations in Banach spaces , 1988 .

[15]  A. I. Subbotin.  Generalization of the main equation of differential game theory , 1984 .

[16]  V. Barbu.  Hamilton-Jacobi equations and nonlinear control problems , 1986 .

[17]  N. Lukoyanov,et al.  Hamilton–Jacobi functional equations and differential games for neutral-type systems , 2017 .

[18]  I. P. Natanson,et al.  Theory of Functions of a Real Variable , 1955 .

[19]  A. M. Zverkin,et al.  DIFFERENTIAL EQUATIONS WITH A PERTURBED ARGUMENT , 1962 .

[20]  Jianjun Zhou,et al.  Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations , 2019, Int. J. Control.

[21]  E. Bayraktar,et al.  Path-dependent Hamilton–Jacobi equations in infinite dimensions , 2017, Journal of Functional Analysis.

[22]  P. Lions,et al.  Hamilton-Jacobi equations in infinite dimensions, III , 1986 .

[23]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[24]  A. V. Kim,et al.  Functional Differential Equations: Application of i-smooth calculus , 1999 .

[25]  M. Crandall,et al.  Hamilton-Jacobi Equations in Infinite Dimensions , 2003 .

[26]  Yuri F. Saporito.  Stochastic Control and Differential Games with Path-Dependent Influence of Controls on Dynamics and Running Cost , 2019, SIAM J. Control. Optim..

[27]  N. Lukoyanov.  FUNCTIONAL HAMILTON-JACOBI TYPE EQUATIONS IN CI-DERIVATIVES FOR SYSTEMS WITH DISTRIBUTED DELAYS , 2016 .

[28]  X. Gong.  Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems , 2007 .

[29]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[30]  N. Lukoyanov.  On optimality conditions for the guaranteed result in control problems for time-delay systems , 2010 .

[31]  P. Lions,et al.  Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions , 1986 .

[32]  P. Wolenski.  Hamilton-Jacobi theory for hereditary control problems , 1994 .

[33]  Minimax Solutions of Functional Equations of the Hamilton–Jacobi Type for Hereditary Systems , 2001 .

[34]  N. Lukoyanov.  A Hamilton-Jacobi type equation in control problems with hereditary information , 2000 .

[35]  Bruno Dupire,et al.  Functional Itô Calculus , 2009 .

[36]  P. Lions,et al.  Hamilton-Jacobi equations in infinite dimensions I. Uniqueness of viscosity solutions , 1985 .

[37]  Hiroshi Ito,et al.  On Saturation, Discontinuities, and Delays, in iISS and ISS Feedback Control Redesign , 2012, IEEE Transactions on Automatic Control.

[38]  Jianfeng Zhang,et al.  Two Person Zero-Sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation , 2012, SIAM J. Control. Optim..

[39]  A. V. Kim.  Functional Differential Equations , 1999 .

[40]  Piermarco Cannarsa,et al.  Value function and optimality conditions for semilinear control problems , 1992 .

[41]  Harvey Thomas Banks,et al.  Application of abstract variational theory to hereditary systems--A survey , 1974 .

[42]  Andrei I. Subbotin,et al.  Generalized solutions of first-order PDEs - the dynamical optimization perspective , 1994, Systems and control.

[43]  N. Lukoyanov.  Minimax and viscosity solutions in optimization problems for hereditary systems , 2010 .

[44]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[45]  E. Barron,et al.  The necessary conditions for optimal control in Hilbert spaces , 1988 .

[46]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[47]  M. Namik Oguztöreli,et al.  Time-lag control systems , 1966 .

[48]  J.-P. Aubin,et al.  History Path Dependent Optimal Control and Portfolio Valuation and Management , 2002 .