The approximation of reachable sets of control systems with integral constraint on controls

Abstract

Abstract. In this paper an approximation method for the construction of reachable sets of control systems with integral constraints on the control is considered. It is assumed that the control system is non-linear with respect to the phase state vector and is linear with respect to the control vector. The admissible control functions are chosen from the ball centered at the origin with radius µ0 in Lp, p > 1. The reachable set is replaced by the set which consists of finite number of points. The estimated accuracy of the Hausdorff distance between the reachable set and the set which is approximately constructed is obtained.2000 Mathematics Subject Classification: Primary 34A34; Secondary 93B03.

Approximation of the Sections of the Set of Trajectories of the Control System Described by a Nonlinear Volterra Integral Equation

“…The convergence of the approximations is proved. Using the algorithm presented in [11], which allows to align the finite number of piecewise constant control functions from the set of controls U H(ε),Γ,Λ,σ p,µ , and an arbitrary numerical method for calculation of the solution of nonlinear Volterra type integral equation, it is possible to carry out calculation of the section X H(ε),Γ,Λ,σ p,µ ξ . The approximate construction of the sections permits the solution of various types of optimal control problems arising in different fields of theory and applications.…”

“…The same problems for the systems described by a Volterra type integral equation are considered in papers [1,2,3,21,25,29] (see, also the references in these papers). Various properties, including numerical methods for construction of the attainable sets of the control systems described by an ordinary differential equations with integral constraints on the controls are studied in papers [5,7,10,11,16,19,20,24]. Note that integral constraint on the controls is essentially different than geometric constraint.…”

An Algorithm for Computing Boundary Points of Reachable Sets of Control Systems Under Integral Constraints

“…The calculations of boundary points require the solution of the system of equations 5) and also the integration of system (2.1) with zeros of system (2.5) as the initial points for (2.1). In case m = n the system (2.5) consists of a single equation ψ 0 (q) = 0.…”

“…The numerical algorithms for constructing approximations of reachable sets of control systems were investigated in many works (see, for example [2, 4, 7, 9-12, 14, 15, 17]). The properties of reachable sets under integral constraints and algorithms for their construction were studied in [1,5,6,16]. For systems with pointwise constraints on the control it is known (see, for example, [13]) that the control, which steers the trajectory to the boundary of the reachable set, satisfies the Pontryagin maximum principle.…”

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

“…Integral constraint on the control functions is inevitable if the control effort is exhausted by consumption. 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].…”