Approximation of the attainable sets of the nonlinear control systems with integral constraint on controls

On the existence of ε-optimal trajectories of the control systems with constrained control resources

“…Integral constraint on the control functions is inevitable if the control resource is exhausted by consumption, such as energy, fuel, food and …nance (see, e.g. [5], [14], [16], [17]). …”

“…In [7] the approximation of 76 ANAR HUSEYIN the sets of trajectories of the aforementioned systems is discussed. A similar problem for the systems described by the ordinary di¤erential equations is considered in [5]. Existence of optimal controls and controllability of the systems described by the Urysohn type integral equations are discussed in [2], [12] where it is assumed that control functions have a geometric constraint.…”

Compactness of the Set of Trajectories of the Control System Described by a Urysohn Type Integral Equation

“…Such controls arise in various problems of economics, medicine, biology, mechanics and physics (see, [8] - [11]). Note that control system with integral constraint on the control functions, where the behavior of the system is given by a nonlinear differential equation is investigated in [8,9].…”

Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra integral equation

“…The control systems with integral constraint on controls are generally needed in modelling the systems having limited energy resources which are exhausted by consumption, such as fuel or finance (see, e.g. [2], [3], [4], [5], [7], [9]). For example, the motion of a flying apparatus with variable mass is described in the form of a control system, where the control functions have integral constraint (see e.g.…”

“…The background of the continuity of the set of trajectories with respect to p is a theorem proved in [5] which asserts that the closed balls of the space L p (p > 1) with radius µ and centered at the origin are continuous with respect to p in Hausdorff metric. In [3], an approximation method for the construction of attainable sets of a control system with integral constraint on the controls is given, where it is assumed that the dynamics of the system is described by a nonlinear ordinary differential equation. Precompactness of the set of trajectories is discussed in [7], where the behavior of the system is described by a nonlinear Volterra integral equation.…”