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Families of exponentials. The method of moments in controllability problems for distributed parameter systems. (English) Zbl 0866.93001

Cambridge: Cambridge Univ. Press. xv, 302 p. (1995).
The moment problem associated with a sequence of functions \(\{p_n(\xi)\}\) in a measurable set \(K\subseteq\mathbb{R}^n\) is: given a sequence \(\{c_n\}\subseteq\mathbb{R}\), find a function \(f(\xi)\) such that \[ \int_Kp_n(\xi)f(\xi)d\xi=c_n\qquad (n=1,2,\dots)\tag{1} \] (the name is often used for the more general problem where \(f(\xi)d\xi\) is replaced by \(\mu(d\xi)\), \(\mu\) a measure in \(K\)). One way to solve (1) (when it works) is to find a biorthogonal sequence \(\{q_n(\xi)\}\), i.e. a sequence satisfying \(\int_Kp_m(\xi)q_n(\xi)d\xi=\delta_{mn}\); then \(f(\xi)=\sum c_nq_n(\xi)\) is (formally) a solution. Already sporting many applications to analysis and probability, the moment problem found fresh ones in the early sixties with the birth of control theory of partial differential equations. To fix ideas, let \({\mathcal A}=\sum\sum\partial/\partial x_j(a_{jk}(x)\partial/\partial x_k)+c(x)\) be a selfadjoint uniformly elliptic operator in a domain \(\Omega\subseteq\mathbb{R}^n\) with an homogeneous boundary condition on the boundary \(\Gamma\). Consider the controlled parabolic differential equation \[ y_t(t,x)={\mathcal A}y(t,x)+g(x)u(t)\quad (0\leq t\leq T),\quad y(0,x)=0,\tag{2} \] where the scalar function \(u(t)\) is the control. The controllability problem of driving the solution to a target \(\overline y(x)\) (that is, of achieving the final condition \(y(T,x)=\overline y(x)\)) reduces to the moment problem (1) for \(K=[0,T]\), \(p_n(t)=e^{-\lambda_n(T-t)}(\{-\lambda_n\}\) the eigenvalues of \(\mathcal A\)) and \(c_n=\overline y_n/g_n\), \(\overline y_n\) (resp. \(g_n\)) the Fourier coefficients of \(\overline y(x)\) (resp. of \(g(x)\)) with respect to the eigenfunctions \(\{\phi_n\}\) (the problem of driving an initial condition \(y_0(x)\) to equilibrium \(y(T,x)=0\) is solved in the same way). The corresponding controllability problem for the hyperbolic equation \[ y_{tt}(t,x)={\mathcal A}y(t,x)+g(x)u(t)\quad (0\leq t\leq T),\quad y(0,x)=0,\quad y_t(0,x)=0\tag{3} \] takes two target conditions \(y(T,x)=\overline y_0(x)\) and \(y_t(T,x)=\overline y_1(x)\) and can be likewise reduced to a moment problem in \(K=[0,T]\), this time with \(p_n(t)=e^{\pm i\sqrt{\lambda_n}(T-t)}\). Among the early advocates of this approach were A. G. Butkovski in his book [Theory of optimal control of distributed parameter systems (Russian), Nauka, Moscow (1965; Zbl 0136.09803)], Yu. V. Egorov [Zhurn. Vychisl. Mat. i Mat. Fiziki 3, 883-904 (1963)] and D. L. Russell [J. Math. Anal. Appl. 18, 542-560 (1967; Zbl 0158.10201)]. At about the same time the reviewer [Commun. Pure Appl. Math. 19, 17-34 (1966; Zbl 0135.36903)] approached the moment problem corresponding to (2) and non-selfadjoint generalizations with two different objectives, one studying uniqueness of the solution (to provide counterexamples to the maximum principle), the other showing that solvability for a given \(\{c_n\}\) does not depend on the interval, which implies time invariance of the set of attainable states. For later developments, see the references in this book. We limit ourselves to mention D. L. Russell [Stud. Appl. Math. 52, 189-211 (1973; Zbl 0274.35041)], the first multidimensional result that does not assume any special symmetry of domains (as do earlier results based, for instance, on separation of variables). Examples can be extended from one-dimensional to finite-dimensional controls, but the theory doesn’t stop there. For an example, consider the following variant of (2), \[ y_t(t,x)={\mathcal A}y(t,x)+\chi(x)u(x,t)\quad(0\leq t\leq T),\quad y(0,x)=0,\tag{4} \] where \(\chi(x)\) is a function supported by a subdomain \(\Omega_0\). Application of the divergence theorem shows that (4) is equivalent to a moment problem (1) with \(K=[0,T]\times\Omega_0\), functions \(p_n(t,x)=\chi(x)\phi_n(x)e^{-\lambda_n(T-t)}\) and, again, \(c_n\) the Fourier coefficients of \(\overline y(x)\).
For many years, information on controllability via moment problems could only be found rummaging in original papers. Except for sporadic partial treatments, the first book fully devoted to the subject was the authors’ monograph [Controllability of distributed parameter systems and families of exponentials (in Russian), UMK BO, Kiev]. Although written by two of the principal actors in the field, popularity of this work was limited chiefly because of the language barrier and unattractive typesetting. The present book is a much improved version of the earlier monograph (in fact, in many senses a totally new work) and it should be useful to most control scholars.
It is divided roughly in two parts: Chapter 1-3 (theory) and Chapters 4-7 (applications). The theory makes contact with many beautiful areas of harmonic analysis, among them abstract moment problems in Hilbert spaces, Hardy spaces, inner and outer operator functions, vector valued singular integrals and various classical problems, such as completeness of sets of exponentials in intervals, all of which is refreshing to learn (or re-learn) with specific applications in mind. The “applications” chapters contain numerous results and references to other methods and works on controllability ranging from the early sixties to the present, among them the application (by Lebeau, Rauch and others) of microlocal analysis to the controllability of hyperbolic equations. Finally, the book comes with an admirably up-to-date bibliography containing for instance the very recent solution by Lebeau and Robbiano (which appeared in 1996) of the boundary controllability problem for parabolic equations when control is applied in an arbitrarily small portion of the boundary, a problem that mystified the pioneers.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
93C25 Control/observation systems in abstract spaces
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