The Semigroup Method for Inverse, Nonlocal, and Nonclassical Problems. Prediction-Control and Prediction-Observation for Evolution Equations: I

摘要

In the present paper, we consider inverse problems for nonautonomous equations, also known as "prediction-control problems." We also consider direct time-nonlocal problems related to the inverse problems. In a number of cases, time-nonlocal problems are treated as prediction-observation problems. In the first part of the paper, we represent, general and special statements of the above-mentioned problems. In the linear case, we introduce the definition of weak and strict (classical) solutions as well as of weak and classical (Hadamard) well-posedness of problem solutions. First, for the linear case, we study problems with a bounded operator generating a semigroup but with a general operator source of a special form. Then we analyze problems for an unbounded operator but in the special case of a scalar source. We note the substantial difference, even in the linear case, of prediction-control and prediction-observation problems from the well-known problems of optimal, programmed, and some other types of control for systems with lumped and distributed parameters. An exposition of abstract differential equations and their applications, semigroup theory, inverse and nonlocal problems, and known problems of control theory can be found in numerous papers (e.g., see [1-20] and the bibliography therein).