Optimal Impulse Control of Dynamical Systems

Using the tools of the Markov Decision Processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. We prove the equivalence of the integral and differential forms of the optimality equation. The theory is illustrated by an example from mathematical epidemiology. The developed methods can be also useful for the study of piecewise deterministic Markov processes.

[1]  Onésimo Hernández-Lerma,et al.  Markov Control Processes , 1996 .

[3]  Oussama Habachi,et al.  Infinite horizon optimal impulsive control with applications to Internet congestion control , 2015, Int. J. Control.

[4]  O. Costa,et al.  Impulse and continuous control of piecewise deterministic Markov processes , 2000 .

[5]  F. Dufour,et al.  Impulsive Control for Continuous-Time Markov Decision Processes: A Linear Programming Approach , 2014, Applied Mathematics & Optimization.

[6]  O. Hernández-Lerma,et al.  Further topics on discrete-time Markov control processes , 1999 .

[7]  A. A. Yushkevich,et al.  Verification Theorems for Markov Decision Processes with Controlled Deterministic Drift and Gradual and Impulsive Controls , 1990 .

[8]  F. Dufour,et al.  Optimal impulsive control of piecewise deterministic Markov processes , 2016 .

[9]  Hailiang Yang,et al.  Classical and Impulse Control for the Optimization of Dividend and Proportional Reinsurance Policies with Regime Switching , 2010, J. Optim. Theory Appl..

[10]  M. K rn,et al.  Stochastic Optimal Control , 1988 .

[11]  Winfried Gleiβner The spread of epidemics , 1988 .

[12]  Alexei B. Piunovskiy,et al.  An explicit optimal isolation policy for a deterministic epidemic model , 2005, Appl. Math. Comput..

[13]  Maurice Robin,et al.  On Some Impulse Control Problems with Constraint , 2017, SIAM J. Control. Optim..

[14]  A. Piunovskiy,et al.  An explicit optimal intervention policy for a deterministic epidemic model , 2008 .

[15]  Vladimir Protopopescu,et al.  Optimal control of continuous systems with impulse controls , 2015 .

[16]  Phil Howlett,et al.  Stochastic Optimal Control of a Solar Car , 2001 .

[17]  K. H. Wong,et al.  Optimal Impulsive Control Problem with Application to Human Immunodeficiency Virus Treatment , 2011, J. Optim. Theory Appl..

[18]  Jane J. Ye,et al.  Impulse Control of Piecewise Deterministic Markov Processes , 1995 .

[19]  Benoîte de Saporta,et al.  Optimal strategies for impulse control of piecewise deterministic Markov processes , 2017, Autom..

[20]  Mihail Zervos,et al.  A singular control model with application to the goodwill problem , 2007, 0711.2143.

[21]  A. Abakuks An optimal isolation policy for an epidemic , 1973, Journal of Applied Probability.

[22]  Denis Arzelier,et al.  Measures and LMIs for Impulsive Nonlinear Optimal Control , 2014, IEEE Transactions on Automatic Control.