Control Of Hyperbolic Systems Research Articles

Variational optimality condition in control of hyperbolic systems with boundary delay parameters

An optimal control problem of a first-order hyperbolic system is studied, in which a boundary condition at one of the ends is determined from a controlled system of ordinary differential equations with constant state lag. Control functions are bounded and measurable functions. The system of ordinary differential equations at the boundary is linear in state. However the matrix of coefficients depends on control functions. Therefore, the optimality condition of Pontryagin’s maximum principle type in this problem is a necessary, but not a sufficient optimality condition. In this paper, the problem is reduced to an optimal control problem of a special system of ordinary differential equations. The proposed approach is based on the use of an exact formula of the cost functional increment. The reduced problem can be solved using a wide range of effective methods used for optimization problems in systems of ordinary differential equations. Problems of this kind arise when modeling thermal separation processes, suppression of mechanical vibrations in drilling, wave processes and population dynamics.

Feedback boundary control of linear hyperbolic equations with stiff source term

ABSTRACTWe consider feedback boundary control of hyperbolic systems with stiff source terms. By combining weighted Lyapunov functions, the structure is used to derive stabilisation results. In our analysis, we give stabilising feedback laws that allow a robust uniform exponential stabilisation for a whole range of values of an uncertain parameter with a decay rate that is independent of the parameter. In particular, we are interested in the limit case when the relaxation parameter approaches zero. The result is illustrated with the numerical analysis on the decay rate of the Lyapunov function in terms of the stiff parameter and an application to boundary stabilisation of gas dynamics in pipes.

Reachable Sets of Quasilinear Hyperbolic Control Systems with Applications to Separation Processes

AbstractThis paper is devoted to the analysis of reachable sets for quasilinear distributed parameter systems by using smooth finite dimensional controls. The class of systems under consideration includes models of a moving bed chromatographic process. We propose sufficient conditions on the states of such quasilinear systems to be approximately reachable with a family of controls corresponding to their finite dimensional approximations. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Gradient Controllability for Hyperbolic Systems

This paper deals with the problem of regional gradient controllability of hyp erbolic systems. We show how one can reach a desired state gradient given only on a part of the system evolution domain. Also we explore a numerical approach using Hilbert Uniqueness Method (HUM) that leads to an explicit formula of the optimal control. The obtained results are successfully tested through computer simulations leading to some conjectures.

Linear Operator Inequality and Null Controllability with Vanishing Energy for Unbounded Control Systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $ y_0 \in H$ we associate the minimal “energy” needed to transfer $ y_0 $ to $ 0 $ in a time $ T \ge T_0$ (``energy” of a control being the square of its $ L^2 $ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk [SIAM J. Control Optim., 42 (2003), pp. 1013--1032] for distributed systems. The proofs in the Priola--Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the linear operator inequality.

Optimal control: Nonlocal conditions, computational methods, and the variational principle of maximum

This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton-Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give the survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.

Boundary control for hyperbolic systems involving infinite-order operators with control constraints

In this paper, a class of n × n hyperbolic boundary control problems with control constraints is investigated. We first establish the solvability of an n × n system with non-homogeneous mixed Neumann conditions involving hyperbolic operator of infinite order. Also a boundary control problem for this system is considered. The necessary and sufficient conditions for the control to be optimal are obtained. Two mathematical examples for derived optimality conditions are presented.

Controllability of a fluid in a 1-D tank

We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. The motion of the fluid is assumed to be well-described by the Saint-Venant equations (also called the shallow water equations). We state a local controllability result for this nonlinear control system around every steady state. The main ingredients of the proof are sketched on toy models. We also recall on simple hyperbolic control systems some classical tools to prove the controllability in connection with time-delay control systems.

Time-optimal control of hyperbolic systems with deviating arguments

In this paper the time-optimal control problem for distributed hyperbolic systems in which time delays appear in the integral form both in the state equations and in the boundary conditions is considered. The particular properties of the optimal control are discussed.

Boundary Control of a Class of Hyperbolic Systems

The paper studies a class of symmetric hyperbolic systems with boundary control. We prove the exact controllability of these systems. The exact controllability result is exploited to investigate the associated optimal control problem. For some special cases, we construct explicitly the optimal control law under the feedback form. Our construction illustrates the difficulty that we would meet for the boundary optimal control of hyperbolic systems. This difficulty does not appear if the control operator is bounded as has been observed in the existing literature.

Regional Boundary Controllability of Hyperbolic Systems. Numerical Approach

The problem of regional boundary controllability for hyperbolic sytems is considered. Thus, we show how one can reach a desired state given only on a part of the boundary of the system domain. Also we explore a numerical approach that leads to an explicit formula of the optimal control. The results obtained are successfully tested through computer simulations.

On Construction and Investigation of General Solutions of Problems of Terminal Control for Hyperbolic Systems

On Construction and Investigation of General Solutions of Problems of Terminal Control for Hyperbolic Systems

Controllability of hyperbolic systems under singular perturbations

Controllability of hyperbolic systems under singular perturbations

The Kalman-Popov-Yakubovich Theorem: an overview and new results for hyperbolic control systems

The Kalman-Popov-Yakubovich Theorem: an overview and new results for hyperbolic control systems

Optimal impulse control of hyperbolic systems

The paper studies optimal control of hyperbolic distributed systems by impulse input. Problems of this type arise in many scientific and engineering projects.

Contrôlabilité exacte pour des domaines minces

Yan, J., Controlabilite exacte pour des domaines minces, Asymptotic Analysis 5 (1992) 461-47l. 461 In studying the limit behavior of the operator defined by the method HUM for the exact controllability of hyperbolic systems on a thin domain n X (0, E) in ~n + 1, we observe that as e goes to zero, this operator will not tend to the corresponding operator defined by HUM on the domain n in ~n. We make a correction for the operator on the thin domain so as to get the desired limit behavior.