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Minimal time control problem of a linear heat equation with memory
Abstract
This paper studies a minimal time control problem for a linear heat equation with memory. The purpose of such a problem is to find a control (among certain control constraint set), which steers the solution of the heat equation with memory from a given initial state to a given target as soon as possible. In this paper, we study the existence of optimal control to this problem, show the bang–bang property of the optimal control and build up a necessary and sufficient condition of the optimal control.
... where α and β are two constants. By similar arguments as those in [16] (see the proof of (3.17) in [16]), we can directly check that (H) holds. ...
... where α and β are two constants. By similar arguments as those in [16] (see the proof of (3.17) in [16]), we can directly check that (H) holds. ...
... Such phenomena can be described by partial integro-differential equations (see, for instance, [5], [11] and [18]). There exist many topics on partial integrodifferential equations, such as Pontryagin's maximum principle of optimal control problems, controllability, observability, stability, minimal time control and so on (see, for instance, [3], [2], [4], [8], [9], [19], [14], [15], [1] and [16]). In [16], connections between minimal time and optimal norm control problems for linear heat equations with memory were discussed. ...
This paper studies an equivalence theorem for three different kinds of optimal control problems, which are optimal time control problems, optimal norm control problems, and optimal target control problems. The controlled systems in this paper are internally controlled linear heat equations with memory.
This article is devoted to studying the null controllability of evolution equations with memory terms. The problem is challenging not only because the state equation contains memory terms but also because the classical controllability requirement at the final time has to be reinforced, involving the contribution of the memory term, to ensure that the solution reaches the equilibrium. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the adjoint system. We first consider finite-dimensional dynamical systems involving memory terms and derive rank conditions for controllability. Then the null controllability property is established for some parabolic equations with memory terms, by means of Carleman estimates.
In this paper, minimal time and minimal norm control problems are studied. The target sets considered are the origin of state spaces and controls are point-wisely bounded functions. The system stuided in this paper is assumed to have no the null controllability or the backward uniqueness property. In this study, minimal time and minimal norm control problems depend on two parameters, respectively. Whether these problems hold the bang-bang property also depend on the parameters. We study the bang-bang property for different parameters for minimal time and minimal norm control problems, by assuming some kinds of weak controllability and unique continuation property. These two properties automatically hold for general time-invariant finitely dimensional controlled systems.
We consider the time optimal control problem, with a point target, for a class of infinite dimensional systems with a dynamics governed by an abstract Schrodinger-type equation. The main results establish a Pontryagin-type maximum principle and give sufficient conditions for the bang- bang property of optimal controls. The results are then applied to some systems governed by partial differential equations. The paper ends with a discussion of possible extensions and by stating some open problems.
We prove the equivalence of the minimal time and minimal norm control prob-lems for heat equations on bounded smooth domains of the euclidean space with homogeneous Dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves. We consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time. As a consequence of this equivalence, using the well-known variational characterization of minimal norm controls, we establish necessary and sufficient conditions for the minimal time and the corresponding control.
A modified Fourier’s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.
: A principal technical result of this paper is that the onedimensional heat equation with boundary control is exactly null-controllable with control restricted to an arbitrary set E ae [0; T ] of positive measure. A general abstract argument is presented to show that this implies the `bangbang ' property for time-optimal controls --- i.e., such a control can take only extreme values of (the hull of) the constraint set --- without imposing any condition regarding the target state, in contrast to previous results. Key Words: bang-bang control, nullcontrol, reachability. AMS(MOS) subject classifications (1991): 49K20, 49K30. 1 To appear in SIAM J. Control and Optimization. 2 Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15217; email: hvm09+@andrew.cmu.edui. 3 Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21228; e-mail: hseidman@math.umbc.edui. 1. Introduction Our principal concern will be with the `bang-...
Time optimal control governed by the internally controlled linear Fitzhugh-Nagumo equation with pointwise control constraint is considered. Making use of Ekeland's variational principle, we obtain Pontryagin's maximum principle for a time optimal control problem. Using the maximum principle, the bang-bang property of the optimal controls is established under appropriate assumptions.
In this paper we are concerned with some theoretical questions for FitzHugh-Nagumo equation. First, we recall the system, we briefly explain the meaning of the variables and we present a simple proof of the existence and uniqueness of strong solution. We also consider an optimal control problem for this system.In this context, the goal is to determine how can we act on the system in order to get good properties. We prove the existence of optimal state-contral pairs and,as an application of the Dubovitski-Milyoutin formalism, we deduce the corresponding optimality system. We also connect the optimal control problem with a controllability question and we construct a sequence of controls that produce solutions that converg strongly to desired states. This provides a strategy to make the system behave as desired. Finally, we present some open questions related to the control of this equation.
This paper addresses the study of the integral-type approximate controllability of linear parabolic integro-differential equations. This new controllability is defined by imposing some additional integral-type constraints on the usual approximate controllability and therefore, can be used to keep the state close to the target. The paper is concerned with a special choice of integral kernels, which are multiples of the same exponential function. We reduce the problem of new controllability to the obtention of a unique continuation property for the suitable adjoint system.
Let S be a physical system whose state at any time is described by an n-dimensional vector x(t), where x(t) is determined by a linear differential equation Z = Az, with A a constant matrix. Application of external influences will yield an inhomogeneous equation, Z = Az + f, where f, the 'forcing term', represents the control. A problem of some importance in the theory of control circuits is that of choosing f so as to reduce z to 0 in minimum time. If f is restricted to belong to the class of vectors whose i(th) components can assume only the values =b sub i, the control is said to be of the 'bang-bang' type. The case where all the solutions of Z = Az approach zero as t approaches infinity is considered.
Time optimal control problems for an internally controlled heat equation with pointwise
control constraints are studied. By Pontryagin’s maximum principle and properties of
nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal
control. Using the bang-bang property and establishing certain connections between time
and norm optimal control problems for the heat equation, necessary and sufficient
conditions for the optimal time and the optimal control are obtained.
It is proved that the time-optimal controls associated with arbitrary reachable target temperature distributions in boundary control for the heat equation (with bounds on the admissible controls) are ″bang-bang″ .
In this paper we deal with the null controllability problem for the heat equation with a
memory term by means of boundary controls. For each positive final time T
and when the control is acting on the whole boundary, we prove that there exists a set of
initial conditions such that the null controllability property fails.
In this paper we consider the heat equation with memory in a bounded region Ω ⊂ R d , d ≥ 1, in the case that the propagation speed of the signal is infinite (i.e. the Colemann-Gurtin model). The memory kernel is of class C 1 . We examine its controllability properties both under the action of boundary controls or when the controls are distributed in a subregion of Ω. We prove approximate controllability of the system and, in contrast with this, we prove the existence of initial conditions which cannot be steered to hit the target 0 in a certain time T , of course when the memory kernel is not identically zero. In both the cases we derive our results from well known properties of the heat equation. keyword Heat equation with memory, approximate controllability, control-lability to zero, lack of controllability
In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.
Die Autoren diskutieren neuere Anwendungen des quiprsenzprinzips auf Konstitutionsgleichungen ruhender Wrmeleiter und entwickeln sodann eine Theorie, in welcher die Beeinflussung der freine Energie, der Entropie und des Wrmeflusses durch die Geschichte nicht nur der Temperatur, sondern auch des Temperaturgradienten zugelassen ist. Sie formulieren fr starre Krper dieser Art notwendige und hinreichende Bedingungen fr die Gltigkeit der Clausius-Duhemschen Ungleichung in allen glatten Wrmeprozessen.
In this paper, we study a certain time optimal control problem governed by the internal controlled heat equation and with a closed ball centered at 0 as the target set. We derive, by making use of Pontryagin's maximum principle and a special kind of unique continuation property for solutions of the heat equation, the bang-bang property for time optimal controls of the problem.