arXiv subjects

Valery Y. Glizer

Publications and source records attributed to Valery Y. Glizer.

4 recordsLinked to original sources

arXiv2026

Asymptotic Solution of a Cheap Control Game with Slow and Fast State Variables

A finite-horizon zero-sum linear-quadratic differential game is considered. Its features are: (i) the control cost of the minimizing player in the game's cost functional is much smaller than the control cost of the maximizing player and the state cost; (ii) the cost of the fast state variable in the integrand of the cost functional is a positive semi-definite (but non-zero) quadratic form. These features require developing a significantly novel approach to asymptotic analysis of the matrix Riccati differential equation associated with the considered game. Using this analysis, an asymptotic solution of the game is derived. An illustrative example is presented.

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arXiv2025

Suboptimal open-loop solution of a Stackelberg linear-quadratic differential game with cheap control of a follower: analytical/numerical study

A two-player finite horizon linear-quadratic Stackelberg differential game is considered. The feature of this game is that the control cost of a follower in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. The open-loop solution of this game is studied. Using the game's solvability conditions, obtaining such a game's solution is reduced to the solution of a proper boundary-value problem. Due to the smallness of the follower's control cost, this boundary-value problem is singularly perturbed. The asymptotic behaviour of the solution to this problem is analysed. Based on this analysis, the asymptotic behaviour of the open-loop optimal players' controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players' controls are designed. An illustrative example of a supply chain problem with a small control cost of a retailer is presented.

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arXiv2018

Defender-Attacker-Target Game: Open-Loop Solution

A defender-attacker-target problem with non-moving target is considered. This problem is modeled by a pursuit-evasion zero-sum differential game with linear dynamics and quadratic cost functional. In this game the pursuer is the defender, while the evader is the attacker. The objective of the pursuer is to minimize the cost functional, while the evader has two objectives: to maximize the cost functional and to keep a given terminal state inequality constraint. The open-loop saddle point solution of this game is obtained in the case where the transfer functions of the controllers for the defender and the attacker are of arbitrary orders. Then, this result is applied to the case of the first order controllers for the defender and the attacker. Numerical illustrating examples are presented.

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arXiv2016

Singular Infinite Horizon Quadratic Control of Linear Systems with Known Disturbances: A Regularization Approach

An optimal control problem with an infinite horizon quadratic cost functional for a linear system with a known additive disturbance is considered. The feature of this problem is that a weight matrix of the control cost in the cost functional is singular. Due to this singularity, the problem can be solved neither by application of the Pontriagin's Maximum Principle, nor using the Hamilton-Jacobi-Bellman equation approach, i.e. this problem is singular. Since the weight matrix of the control cost, being singular, is not in general zero, only a part of the control coordinates is singular, while the others are regular. This problem is solved by a regularization method. Namely, it is associated with a new optimal control problem for the same equation of dynamics. The cost functional in this new problem is the sum of the original cost functional and an infinite horizon integral of the squares of the singular control coordinates with a small positive weight. Due to a smallness of this coefficient, the new problem is a partial cheap control problem. Using a perturbation technique, an asymptotic analysis of this partial cheap control problem is carried out. Based on this analysis, the infimum of the cost functional in the original problem is derived, and a minimizing sequence of state-feedback controls is designed. An illustrative example of a singular trajectory tracking is presented.

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