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1

Approximate gradient projection method with general Runge-Kutta schemes and piecewise polynomial controls for optimal control problems

Chryssoverghi I.

Control and Cybernetics

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2005

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Vol. 34, no 2

425-451

EN

This paper addresses the numerical solution of optimal control problems for systems described by ordinary differential equations with control constraints. The state equation is discretized by a general explicit Runge-Kutta scheme and the controls are approximated by functions that are piecewise polynomial, but not necessarily continuous. We then propose an approximate gradient projection method that constructs sequences of discrete controls and progressively refines the discretization. Instead of using the exact discrete cost derivative, which usually requires tedious calculations, we use here an approximate derivative of the cost functional denned by discretizing the continuous adjoint equation by the same Runge-Kutta scheme backward and the integral involved by a Newton-Cotes integration rule, both involving maximal order intermediate approximations. The main result is that strong accumulation points in L2, if they exist, of sequences generated by this method satisfy the weak necessary conditions for optimality for the continuous problem. In the unconstrained case and under additional assumptions, we prove strong convergence in L2 and derive an a posteriori error estimate. Finally, numerical examples are given.

2

Approximate relaxed descent method for optimal control problems

Chryssoverghi I., Coletsos J., Kokkinis B.

Control and Cybernetics

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2001

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Vol. 30, no 4

385-404

EN

We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. Since no convexity assumptions are made on the data, the problem is reformulated in relaxed form. The relaxed state equation is discretized by the implicit trapezoidal scheme and the relaxed controls are approximated by piecewise constant relaxed controls. We then propose a combined descent and discretization method that generates sequences of discrete relaxed controls and progressively refines the discretization. Since here the adjoint of the discrete state equation is not defined, we use, at each iteration, an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation and the integral involved by appropriate trapezoidal schemes. It is proved that accumulation points of sequences constructed by this method satisfy the strong relaxed necessary conditions for optimality for the continuous problem. Finally, the computed relaxed controls can be easily approximated by piecewise constant classical controls.

3

Discrete relaxed method for semilinear parabolic optimal control problem

Chryssoverghi I., Coletsos J., Kokkinis B.

Control and Cybernetics

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1999

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Vol. 28, no 2

157-176

EN

We consider an optimal control problem for systems governed by semilinear parabolic partial differential equations with control and state constraints, without any convexity assumptions. A discrete optimization method is proposed to solve this problem in its relaxed form which combines a penalized Armijo type method with a finite element discretization and constructs sequences of discrete Gamkrelidze relaxed controls. Under appropriate assumptions, we prove that accumulation points of these sequences satisfy the relaxed Pontryagin necessary conditions for optimality. Moreover, we show that the Gamkrelidze controls thus generated can be replaced by simulating piecewise constant classical controls.

4

Discrete approximation of nonconvex hyperbolic optimal control problems with state constraints

Chryssoverghi I., Bacopoulos A., Coletsos J., Kokkinis B.

Control and Cybernetics

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1998

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Vol. 27, no 1

29-50

EN

We consider an opitmal control problem for systems defined by nonlinear hyperbolic partial differential equations with state constraints. Since no convexity assumptions are made on the data, we also consider the control problem in relaxed form. We discretize both the classical and the relaxed problenms by using a finite element method in space and a finite difference scheme in time, the controls being approximated by piecevise constant ones. We develop the existence theory and the necessary conditions for optimality, for the continous and the discrete problems. Finally, we study the behaviour in the limit of discrete optimality, admissibility and extremality properties.