Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence
Valery Y. Glizer Oleg Kelis
Numerical Algebra, Control & Optimization 2017, 7(1): 1-20 doi: 10.3934/naco.2017001

We consider an infinite horizon zero-sum linear-quadratic differential game in the case where the cost functional does not contain a control cost of the minimizing player (the minimizer). This feature means that the game under consideration is singular. For this game, novel definitions of the saddle-point equilibrium and game value are proposed. To obtain these saddle-point equilibrium and game value, we associate the singular game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. Using the solvability conditions, the solution of the cheap control game is reduced to solution of a Riccati matrix algebraic equation with an indefinite quadratic term. This equation is perturbed by a small parameter. Subject to a proper assumption, an asymptotic expansion of a stabilizing solution to this equation is constructed and justified. Using this asymptotic expansion, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.

keywords: Singular differential game infinite horizon zero-sum differential game saddle-point equilibrium game value regularization cheap control game perturbation Riccati matrix algebraic equation asymptotic solution
Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances
Valery Y. Glizer Oleg Kelis
Numerical Algebra, Control & Optimization 2018, 8(2): 211-235 doi: 10.3934/naco.2018013

An infinite horizon quadratic control of a linear system with known disturbance is considered. The feature of the problem is that the cost of some (but in general not all) control coordinates in the cost functional is much smaller than the costs of the other control coordinates and the state cost. Using the control optimality conditions, the solution of this problem is reduced to solution of a hybrid set of three equations, perturbed by a small parameter. One of these equations is a matrix algebraic Riccati equation, while two others are vector and scalar differential equations subject to terminal conditions at infinity. For this set of the equations, a zero-order asymptotic solution is constructed and justified. Using this asymptotic solution, a relation between solutions of the original problem and the problem, obtained from the original one by replacing the small control cost with zero, is established. Based on this relation, the best achievable performance in the original problem is derived. Illustrative examples are presented.

keywords: Infinite horizon partial cheap control system with known disturbance degenerate control problem asymptotic solution of algebraic and differential equations best achievable performance
Statistical process control optimization with variable sampling interval and nonlinear expected loss
Valery Y. Glizer Vladimir Turetsky Emil Bashkansky
Journal of Industrial & Management Optimization 2015, 11(1): 105-133 doi: 10.3934/jimo.2015.11.105
The optimization of a statistical process control with a variable sampling interval is studied, aiming in minimization of the expected loss. This loss is caused by delay in detecting process change and depends nonlinearly on the sampling interval. An approximate solution of this optimization problem is obtained by its decomposition into two simpler subproblems: linear and quadratic. Two approaches to the solution of the quadratic subproblem are proposed. The first approach is based on the Pontryagin's Maximum Principle, leading to an exact analytical solution. The second approach is based on a discretization of the problem and using proper mathematical programming tools, providing an approximate numerical solution. Composite solution of the original problem is constructed. Illustrative examples are presented.
keywords: variable sampling interval nonlinear loss optimal control approximation quadratic programming. Statistical process control optimization

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