ON EXACT NULL CONTROLLABILITY OF DISTRIBUTED SYSTEMS
B. Shklyar
Holon Institute of Technology, Holon, Israel
Introduction
The linear moment problem is defined as follows:
Given sequences and find necessary and sufficient conditions for the existence of linear functional such that
(1)
The moment problem (1) has many important applications. In this paper we present applications of linear moment problem (1) for the investigation of exact null-controllability for linear evolution control equations.
Problem statement
Let be Hilbert spaces, and let be infinitesimal generator of strongly continuous -semigroups in [1]. Consider the abstract evolution control equation
(2)
where is a linear possibly unbounded operator, are Hilbert spaces with continuous dense injections (see [2] for the description of spaces and ).
Let be a mild solution of equation (2) with initial condition
Definition Equation (2) is said to be exact null-controllable on by controls vanishing after time moment if for each there exists a control a.e. on such that
The assumptions
The assumptions on are listed below.
1) The operators has purely point spectrums with no finite limit points. Eigenvalues of have finite multiplicities.
2) There exists such that all mild solutions of the equation are expanded in a series of generalized eigenvectors of the operator converging uniformly for any
Main results
As a preliminary we consider the following:
1) The operator has all the eigenvalues with multiplicity .
2) (one input case). It means that the possibly unbounded operator is defined by an element , i.e. equation (2) can be written in the form
3)
The operator defined by is bounded if and only if
Let the eigenvalues of the operator be enumerated in the order of non-decreasing of absolute values, and let be the eigenvectors of the operator and the adjoint operator respectively.
Denote:
Theorem 1. For equation (2) to be exact null-controllable on by controls vanishing after time moment , it is necessary and sufficient that the following infinite moment problem
(3)
with respect to is solvable for any .
Solution of moment problem (3)
The solvability of moment problem (4) for each essentially depends on the properties of eigenvalues
Definition. The sequence is said to be minimal, if
It is well-known that the sequence is minimal if and only if there exists a sequence biorthogonal to the sequence
Let be the Gram matrix of first elements of above sequence. Denote by the minimal eigenvalue of Each minimal sequence is a linear independent sequence, hence any first elements are linear independent, so . It is easily to show that the sequence decreases, so there exists
Definition. The sequence is said to be strongly minimal, if
Theorem 2. For equation () to be exact null-controllable on by controls vanishing after time moment , it is necessary, that the sequence
is minimal, and sufficient , that:
1) the sequence is strongly minimal,
2)
Solution of moment problem (1) in the case of the normal sequence of the eigenvectors of the operator A
Denote by the maximal eigenvalue of It is easily to show that the sequence increases , so there exists .
Definition The sequence is said to be normal, if and
Theorem 3. Let the sequence of eigenvectors of the operator be a normal sequence
For equation () to be exact null-controllable on by controls vanishing after time moment , it is necessary, that the sequence is minimal,
and sufficient , that it is strongly minimal .
It is well-known [2] that both functional differential linear control systems and linear partial differential control equations of parabolic and hyperbolic type can be written in the form (1), so Theorems 1-3 can be applied for the establishing of exact null controllability conditions for above classes of control distributed systems.
References
1. E. Hille, R. Philips, Functional Analysis and Semi-Groups, AMS, 1957.
2. D. Salamon, Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383–431.
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