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1

Online learning algorithm for zero-sum games with integral reinforcement learning

Vamvoudakis K. G., Vrabie D., Lewis F. L.

Journal of Artificial Intelligence and Soft Computing Research

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2011

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Vol. 1, No. 4

315--332

EN

In this paper we introduce an online algorithm that uses integral reinforcement knowledge for learning the continuous-time zero sum game solution for nonlinear systems with infinite horizon costs and partial knowledge of the system dynamics. This algorithm is a data based approach to the solution of the Hamilton-Jacobi-Isaacs equation and it does not require explicit knowledge on the system’s drift dynamics. A novel adaptive control algorithm is given that is based on policy iteration and implemented using an actor/ disturbance/critic structure having three adaptive approximator structures. All three approximation networks are adapted simultaneously. A persistence of excitation condition is required to guarantee convergence of the critic to the actual optimal value function. Novel adaptive control tuning algorithms are given for critic, disturbance and actor networks. The convergence to the Nash solution of the game is proven, and stability of the system is also guaranteed. Simulation examples support the theoretical result.

2

Convergence of approximate solutions of variational problems

Zaslavski A. J.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1607-1629

EN

We study the structure of approximate solutions of autonomous variational problems on large finite intervals. In our previous research, which was summarized in Zaslavski (2006b), we showed that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data, except in regions close to the endpoints of the time interval. In the present paper we establish convergence of approximate solutions in regions close to the endpoints of the time intervals.

3

On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems

Pickenhain S., Lykina V., Wagner M.

Control and Cybernetics

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2008

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Vol. 37, no 2

451-468

EN

This paper deals with infinite horizon optimal control problems, which are formulated in weighted Sobolev spaces ... [wzór] and weighted Lp-spaces ... [wzór]. We ask for the consequences of the interpretation of the integral within the objective as a Lebesgue or an improper Riemann integral. In order to justify the use of both types of integrals, various applications of infinite horizon problems are presented. We provide examples showing that lower semicontinuity may fail for objectives involving Lebesgue as well as improper Riemann integrals. Further we prove a lower semicontinuity theorem for an objective with Lebesgue integral under more restrictive growth conditions on the integrand.

4

An infinite horizon predictive control algorithm based on multivariable input-output models

Ławryńczuk M., Tatjewski P.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 2

167-180

EN

In this paper an infinite horizon predictive control algorithm, for which closed loop stability is guaranteed, is developed in the framework of multivariable linear input-output models. The original infinite dimensional optimisation problem is transformed into a finite dimensional one with a penalty term. In the unconstrained case the stabilising control law, using a numerically reliable SVD decomposition, is derived as an analytical formula, calculated off-line. Considering constraints needs solving on-line a quadratic programming problem. Additionally, it is shown how free and forced responses can be calculated without the necessity of solving a matrix Diophantine equation.

5

Anoccupational measure solution to a singularly perturbed optimal control problem

Artstein Z.

Control and Cybernetics

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2002

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Vol. 31, no 3

623-642

EN

Employing limit occupational measures, we provide an explicit solution to a singularly perturbed optimal control problem for which the order reduction method does not apply.

6

Infinite horizon reflected backward stochastic differential equations and applications in mixed control and game problems

Hamadène S., Lepeltier J.-P., Wu Z.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 2

211--234

EN

We prove existence and uniqueness results of the solution for infinite horizon reflected backward stochastic differential equations with one or two barriers. We also apply these results to get the existence of optimal control strategy for the mixed control problem and a saddle-point strategy for the mixed game problem when, in both situations, the horizon is infinite.