Approximate construction of attainability sets of control systems with integral constraints on the controls

The approximate construction of attainability sets of control systems with quadratic integral constraints on the controls is considered. It is assumed that a control system is non-linear with respect to the phase variable and linear with respect to the variable which describes the controlling action. The approximation of the attainability sets of a control system is accomplished in several stages. The latter class of controls generates a finite number of trajectories of the system. The trajectories of the system are then replaced by Euler broken lines. An estimate of the accuracy of the Hausdorff distance between the attainability set and the set which has been approximately constructed is obtained.     

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

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 L_p, 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.     

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

In this paper the control system with limited control resources is studied, where the behavior of the system is described by a nonlinear Volterra integral equation. The admissible control functions are chosen from the closed ball centered at the origin with radius µ in L _p (p > 1). It is proved that the set of trajectories generated by all admissible control functions is Lipschitz continuous with respect to µ for each fixed p, and is continuous with respect to p for each fixed µ. An upper estimate for the diameter of the set of trajectories is given.     

Linear-quadratic control and quadratic differential forms for multidimensional behaviors

This paper deals with systems described by constant coefficient linear partial differential equations (nD-systems) from a behavioral point of view. In this context we treat the linear quadratic control problem where the performance functional is the integral of a quadratic differential form. We look for characterizations of the set of stationary trajectories and of the set of local minimal trajectories with respect to compact support variations, turning out that they are equal if the system is dissipative. Finally we provide conditions for regular implementability of this set of trajectories and give an explicit representation of an optimal controller.     

Solution of an optimal control problem with phase constraints by the double variation method

An optimal control problem with an integral quality index specified in a finite time interval is formulated for a model of economic growth that leads to emission of greenhouse gases. The controlled system is linear with respect to control. The problem contains phase constraints that abandon emission of greenhouse gases above some predefined time-dependent limit. As is known, optimal control problems with phase constraints fall beyond the sphere of efficient application of the Pontryagin maximum principle because, for such problems, this principle is formulated in a complicated form difficult for analytic treatment in particular situations. In this study, the analytic structure of the optimal control and phase trajectories is constructed using the double variation method.     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.     

Bounds on the number of switchings for trajectories of piecewise analytic vector fields

We study the trajectories of systems $\text{x}\text{?}\text{= X(x)}$, where X is a continuous “extendably piecewise analytic” vector field, i.e., a continuous vector field X such that the domain of ? admits a locally finite partition $\text{I}$ into sets such that for each A ∈ $\text{I}$ there is a vector field X_A which is analytic on a neighborhood of the closure of A and whose restriction to A coincides with that of X. We prove that the trajectories are piecewise analytic, with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number T. This result implies the existence of a regular synthesis for optimal control problems with a strictly convex Lagrangian, and a linear dynamics with polyhedral constraints on the controls.     

Hypercubes are minimal controlled invariants for discrete-time linear systems with quantized scalar input

Quantized linear systems are a widely studied class of nonlinear dynamics resulting from the control of a linear system through finite inputs. The stabilization problem for these models shall be studied in terms of the so-called practical stability notion that essentially consists in confining the trajectories into sufficiently small neighborhoods of the equilibrium (ultimate boundedness).We study the problem of describing the smallest sets into which any feedback can ultimately confine the state, for a given linear single-input system with an assigned finite set of admissible input values (quantization). We show that the family of hypercubes in canonical controller form contains a controlled invariant set of minimal size. A comparison is presented which quantifies the improvement in tightness of the analysis technique based on hypercubes with respect to classical results using quadratic Lyapunov functions.     

The closure of the sheaf of trajectories of a linear control system with integral constraints

We consider a linear system with discontinuous coefficients controlled by a parameter under an integral constraint imposed on the control resource. It is well known that in such problems the closure of the sheaf of trajectories that correspond to ordinary controls (piecewise constant or measurable functions) coincides with the sheaf of trajectories in a generalized problem, where for generalized controls one uses finite additive measures of bounded variation. Therewith the closure is defined in the topology of pointwise convergence, because the limit elements (the generalized trajectories) may be discontinuous functions. In this paper we prove that any generalized trajectory can be approximated by a sequence of ordinary solutions to the initial system. We propose a concrete technique for constructing such sequences.     

The number of cells of a dynamical system

In a dynamical system with a finite number of elementary stationary points, in which just these points serve as the limiting sets of its trajectories, a component of the connection of the set of trajectory points with the common positive and common negative limiting set is called a cell. An example is constructed which shows that a dynamical system can have any finite number of cells even though the number of stationary points is fixed.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 707–714, June, 1968.     

Conditions for unique solvability of inverse problems for controllable dynamical systems

Problems arising in the applications of control theory of dynamical systems, when part of the variables of the mathematical model of the system is unknown and is subject to determination from information on the output of the system, are considered. Fundamental among them are the observation and identification problems, when unknown are the state of the system and its parameters, respectively, and also the problem of inverting the system, in which one seeks the control. Based on an analysis of mappings that are generated by an extended measurement vector, conditions for the unique solvability of the afore-mentioned problems with respect to a single trajectory or a set of trajectories are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1359–1366, October, 1992.     

Discontinuous Solutions to Unbounded Differential Inclusions under State Constraints. Applications to Optimal Control Problems

We consider a nonconvex and unbounded differential inclusion derived from a control system whose control sets are time and space-dependent. We extend the inclusion in order to allow discontinuous trajectories. We prove that the set of solutions of the original inclusion is dense in the set of solutions of the extended inclusion and, moreover, these last solutions are stable with respect to the initial data. Both of these results are also proven in the presence of state and integral constraints (assuming suitable conditions at the boundary of the constraining set). As an application, the value function of a Mayer problem is shown to be continuous and the unique viscosity solution of a Hamilton–Jacobi equation with suitable boundary conditions.     

Modelling of the controlled motion ofa point on a plane

A solution of the problem of feedback control of the motion of a point on a plane is presented. The equations of the controlprogramme (the objective) are set up as a system of differential equations with a given set of singular trajectories in the domain of admissible positions of the controlled point, as well as a given topological structure of the partition into trajectories. These equations define the vector field of velocities of the programmed motions of the point and are used to find the corresponding control forces.     

The Canonical Theory of the Impulse Process Optimality

The paper is devoted to the development of the canonical theory of the Hamilton–Jacobi optimality for nonlinear dynamical systems with controls of the vector measure type and with trajectories of bounded variation. Infinitesimal conditions of the strong and weak monotonicity of continuous Lyapunov-type functions with respect to the impulsive dynamical system are formulated. Necessary and sufficient conditions of the global optimality for the problem of the optimal impulsive control with general end restrictions are represented. The conditions include the sets of weak and strong monotone Lyapunov-type functions and are based on the reduction of the original problem of the optimal impulsive control a finite-dimensional optimization problem on an estimated set of connectable points.     

Determining functionals for random partial differential equations

Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional random dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results.We apply our results to the 2D Navier-Stokes equations forced by a white noise.     

On the problem of output feedback control synthesis under uncertainty and finite-time observers

We consider the problem of controlling a linear system of ordinary differential equations with a linear observable output. The system contains uncertain items (disturbances), for which we know only “hard” pointwise constraints. The problem of synthesizing a control that brings the trajectories of the system into a given target set in finite time is solved under weakened conditions without assuming that the control and the disturbance are of the same type. To this end, we suggest an approach that amounts to constructing an information set and a weakly invariant set with subsequent “aiming” of the first set at the second. Both stages are carried out in a finite-dimensional space, which permits one to use an efficient algorithm for solving the synthesis problem approximately on the basis of the ellipsoidal calculus technique. The results are illustrated by an example in which the control of a linear oscillation system is constructed.     

On the representation of switched systems with inputs by perturbed control systems

This paper provides representations of switched systems described by controlled differential inclusions, in terms of perturbed control systems. The control systems have dynamics given by differential equations, and their inputs consist of the original controls together with disturbances that evolve in compact sets; their sets of maximal trajectories contain, as a dense subset, the set of maximal trajectories of the original system. Several applications to control theory, dealing with properties of stability with respect to inputs and of detectability, are derived as a consequence of the representation theorem.     

Solution of one of Birkhoff''s problems : PMM vol. 42, no. 6, 1978, pp. 1136–1141

The criterion of existence of an integral linear with respect to generalized velocities over the set of trajectories with the same total energy is proved for conservative dynamic systems with two degrees of freedom.     

On the space of measurable functions and its topology determined by the Choquet integral

Let (X,A,μ) be a finite nonadditive measure space and M be the set of all finite measurable functions on X. The topology on M, which is determined by the Choquet integral with respect to μ, is investigated. The relationship between this topology and the one determined by the Sugeno integral is examined. Some interesting examples are included.     

One control problem of a pencil of trajectories

A linear control problemwith a nonsingleton initial set is dealt with. For this problem,we consider the problem of transforming this initial set along trajectories of a linear controlled system into some fixed terminal set in a finite time using a single control. Various aspects of this control problem are studied using the machinery of support functions.