Inverse problems of dynamics of controllable systems with distributed parameters (Q1921706)

Inverse problems are perhaps more important in engineering and applied physics than the so called direct problems. In a typical classroom problem one assumes that the behavior of a dynamical system is known and predicts the output of the system for a given input. In many real-life problems, the properties of the system are not known and experiments may be carried out to find some characteristics of the system by relating its output to the input. Thus, whole classes of problems may be defined. For example, let the state of the system be given by \(y=y(t)\) on some time interval \(t\in T\). An admissible control \(u(t)\) acts on the system which evolves according to some law \(y(\cdot,t)=A(u(\cdot,t))\). Here, \(A\) is a Volterra operator (or one with the Volterra property). This means that only past and present actions of the control \(u\) affect the behavior of the system. In other words, there is no ``peaking into the future''. Since most control and design problems may be classified as inverse problems, a huge amount has been written on this broad subject. The author proceeds to characterize certain classes of such problems. Let the initial state be known \(y(0)= y_0\). A class of feasible controls is known. Controls could be either inputs into the system deliberately applied, or some perturbations such as noise, wind, which some authors may call uncontrollable inputs. The system evolves according to some law \(y(\cdot)= Au(\cdot)\). Observation, or measurement of the system, is described by the observation operator \(C\): \(z=Cy\), or \(z=CAu\), and the control or input may be determined by the measurement \(u=Bz\). If the operator \(CA\) is only approximately known, or guessed, several techniques exist for minimizing the discrepancy functional: \(J(u)=|z(\cdot)-CAu(\cdot)|\). Most popular is the least squares technique. However, some serious problems arise at this point. Stability of this procedure should be questioned. But even if the problem is well posed, to an approximate measurement \(z\) there may correspond a control that is unacceptable for other reasons. It may not be feasible, or it may be far from optimal according to other optimality criteria assigned to the system. In general, the operator \(B\) is multivalent, and the whole problem of optimal control and minimal discrepancy may be ill-posed. A theory of ill-posed problems was developed by A. N. Tikhonov in the 1960s. The Tikhonov ideas were described in an excellent summary by \textit{V. Ya. Arsenin} [Russ. Math. Surv. 31, No. 6, 93-107 (1976); translation from Usp. Mat. Nauk 31, No. 6, 89-101 (1976; Zbl 0347.93025)]. Subsequently, several monographs have been published [e.g., \textit{A. N. Tikhonov} and \textit{V. Ya. Arsenin}, Methods for the solution of ill-posed problems, Nauka Moscow (1979; Zbl 0499.65030)]. Also much was accomplished in related topics by N. N. Krasovskij and his collaborators. After a brief, but very clear exposition of inverse problems of dynamics the author proceeds to discuss restoration of desirable characteristics both in ``real time'' (i.e. as they occur), or in ``floating time''. He favors the techniques of the Krasovskij school used for inverse problems of game theory. The following principles are stated. 1. Algorithms must be of finite steps. Measurements made at such finite time nodes must be processed during the intervals between the nodes. 2. Algorithms must be regularizing. 3. ``Real time'' tempo of restoration is required. 4. They must possess Volterra's property. (Must not ``peak into the future''.) 5. Must be simple. The author offers four examples illustrating his exposition. They include restoration of thermophysical characteristics for a thin elastic bar, restoring mechanical properties of an elastic medium, characteristics of a chemical reactor and finally restoration of sources of disturbance in a medium, with dynamics of the process described by \[ y_{tt}= Ay+q(t,x)_{\chi_G}+ f(t,x),\qquad x\in\Omega, \] with appropriate boundary and initial conditions (\(t\in T\); \(x\in G(t)\); \(\chi\) is the characteristic function and describes the location of the disturbance). This is a well-written, largely expository article. The reviewer recommends it to all graduate students interested in control theory or related topics.