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A Bolza optimal synthesis problem for singular estimate control systems

Lasiecka I., Tuffaha A.

Control and Cybernetics

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2009

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Vol. 38, no 4B

1429-1460

EN

Bolza problem governed by PDE control systems with unbounded controls is considered. The motivating example is fluid structure interaction model with boundary-interface controls. The aim of the work is to provide optimal feedback synthesis associated with well denned gain operator constructed from the Riccati equation. The dynamics considered is of mixed parabolic-hyperbolic type which prevents applicability of tools developed earlier for analytic semigroups. It is shown, however, that the control operator along with the generator of the semigroup under consideration satisfy singular estimate referred to as Revisited Singular Estimate (RSE). This estimate, which measures "unboundedness" of control actions, is a generalization and a weaker form of Singular Estimate (SE) treated in the past literature. The main result of the paper provides Riccati theory developed for this new class of control systems labeled as RSECS (Revisited Singular Estimate Control Systems). The important feature is that the gain operator, constructed via Riccati operator, is consistent with the optimal feedback synthesis. The gain operator, though unbounded, has a controlled algebraically singularity at the terminal point. This enables one to establish well-posedness of the Riccati solutions and of the optimal feedback representation. An application of the theoretical framework to boundary control of a fluid-structure interaction model is given.

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A Method for Constructing ε-value Functions for The Bolza Problem of Optimal Control

Pustelnik J.

International Journal of Applied Mathematics and Computer Science

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2005

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Vol. 15, no 2

177-186

EN

The problem considered is that of approximate minimisation of the Bolza problem of optimal control. Starting from Bellman's method of dynamic programming, we define the ε-value function to be an approximation to the value function being a solution to the Hamilton-Jacobi equation. The paper shows an approach that can be used to construct an algorithm for calculating the values of an ε-value function at given points, thus approximating the respective values of the value function.

3

A general statement of dual first-order sufficient optimality conditions for the generalized problem of Bolza

Galewska E.

Demonstratio Mathematica

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2003

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Vol. 36, nr 3

711--720

EN

In this paper we provide first-order sufficient optimality conditions for the generalized problem of Bolza when all arcs take values in a separable Hilbert space. Our approach consists in the explicit construction of a quadratic function that satisfies the dual Hamilton-Jacobi inequality. The essential role in the generalized conditions plays the existence of a certain function for which a certain inequality holds.

4

An Algorithm for Construction of varepsilon-Value Functions for the Bolza Control Problem

Jacewicz E.

International Journal of Applied Mathematics and Computer Science

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2001

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Vol. 11, no 2

391-428

EN

The problem considered is that of approximate numerical minimisation of the non-linear control problem of Bolza. Starting from the classical dynamic programming method of Bellman, an varepsilon-value function is defined as an approximation for the value function being a solution to the Hamilton-Jacobi equation. The paper shows how an varepsilon-value function which maintains suitable properties analogous to the original Hamilton-Jacobi value function can be constructed using a stable numerical algorithm. The paper shows the numerical closeness of the approximate minimum to the infimum of the Bolza functional.