The problem of the optimal control of a hybrid system (HS), the continuous motion of which alternates with discrete changes (switchings) in which the state space changes, is considered. The change in the dimension of the state space occurs, for example, when the number of controlled objects changes, which is typical, in particular, for the problems of controlling groups of moving objects of variable compositions. The switching times are not predefined. They are determined as a result of minimizing the functional, while processes with instantaneous multiple switchings are not excluded. The necessary conditions for the optimality of the control of such systems are proved. Due to the presence of instantaneous multiple switchings, these conditions differ from traditional ones, in particular, by the equations for auxiliary variables. The application of optimality conditions is demonstrated by an academic example.

We consider the problem of trajectory optimization by a switchable system whose continuous motion is described by differential equations; and discrete state changes (switching), by recurrent inclusions. Its motion is continuously controlled by choosing the state of the discrete part of the system. The number of switchings and time of switching are not predefined. The quality of the trajectory is characterized by a functional that takes into account the costs of each switch. Together with the task of optimizing the trajectories of motion, the task of finding the minimum number of switchings at which the value of the quality functional does not exceed the given value is solved.

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The linear-quadratic problem of synthesis optimal control of switched systems is considered. Continuous change of state of the system is described by linear differential equations, and instantaneous discrete changes of state (switching) – linear recurrent equations. The moments of switching, and their number is not prespecified. The quality of control is characterized by a quadratic functional, which takes into account the cost of each switch. The considered problem generalizes the classical linear-quadratic problems of optimal control of continuous, discrete and continuous-discrete systems, transferring them to a new class of dynamic systems – switchable (hybrid) control systems. Together with the problem of optimal control synthesis, the problem of minimizing the number of switchings, characteristic of hybrid systems, is relevant. The peculiarity of the synthesis of optimal switchable systems is that the price function in the considered problem is not quadratic. Therefore, it is proposed to build a price function from auxiliary, so-called price moment functions, each of which is defined as the minimum value of the quality functional at fixed switching moments and is quadratic. At the same time, the optimal positional control, linear in state, depends nonlinearly on switching moments. Optimization of these moments becomes the last stage of the synthesis. The proposed computer-aided synthesis technology makes it possible to find the optimal “controlling complex”, including the number of switches, the switching moments, as well as the control of continuous and discrete movements of the system. The application of the developed technology is demonstrated on an academic example of synthesis.