Time-optimal control of nonlinear parabolic systems with constrained derivative of control, existence theorem

The studies of the control problems for an evolution system with a bounded derivative ( = the derivative with respect to time) of the control were stimulated by solving practical problems. Our problem is to find an optimal control of a thermal process where the control acts by means of boundary conditions. In more details, an iron body to be heated up is placed into a furnace which temperature is considered to be everywhere constant at each fixed time. The furnace temperature may change within time according to a plan determined in advance, but the speed of the temperature changes, i.e. the derivative of the control, is bounded due to construction parameters of the furnace. Our aim is to heat the body up in a minimum time and, at the same time, the furnace-temperature changes must not exceed the maximal possible speed, and also certain constraints on the temperature inside the body ( = the state-space constraints) must be fulfilled, namely an effective thermoelastic stress must not exceed the critical level prescribed in advance. However, in this paper the state-space constraints will be considered only in a general manner and the problem is thus reduced to a nonlinear parabolic system without any consideration of the quasistationary elliptic problem arising from the elastic-stress equation. Moreover, also the heat transfer operator is considered only in an abstract manner. Generally speaking, in practical situations the derivative of the control is, in fact, constrained very often in consequence of various construction reasons, but mostly the changes of the control may be far quicker than the changes of the state in the controlled system. In such a case it is highly apposite to admit the controls of the bang-