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The paper deals with an optimal control problem for the simplest version of a “reduced” linear impulsive continuity equation. The latter is introduced in our recent papers as a model of dynamical ensembles enduring jumps, or impulsive ODEs having a probabilistic uncertainty in the initial data. The model, addressed in the manuscript, is, in fact, equivalent to the mentioned impulsive one, while it is stated within the usual, continuous setup. The price for this reduction is the appearance of an integral constraint on control, which makes the problem non-standard.

The paper is concerned with a nonlinear optimal impulsive control problem with trajectories of bounded variation. Vector Lebesgue-Stieltjes measures play the part of controls. Studied were state and mixed constraints in the conventional and fast times, as well as joint conditions for trajectory and impulsive control. A method of reduction to the classical problem of optimal control was developed relying on the discontinuous time reparameterization. The original and reduced problems were established to be equivalent.

In this paper, a linear dynamic system is considered under Lp-constraint on control. We establish the existence of the limit shape of reachable sets as time goes to infinity. Asymptotic formulas are obtained for reachable sets and their shapes. The results throw a bridge between the cases of geometric bounds on control and constraints on the total impulse of control, and create a unified picture of the structure of the limit shapes of reachable sets.

We study an optimal control problem for a state-dependent impulse system described by a measure differential equation. A specific time reparameterization technique is developed to reduce the impulsive control problem to the one with bounded controls. Necessary conditions of optimality are obtained by interpreting the Maximum Principle in the reduced problem. An impulsive control improvement scheme is outlined. The results of numeric simulation are presented.

An optimal control problem is considered for a system described by a differential equation with measures; a certain constraint is imposed on the total variation of controlmeasure. Involving the method of discontinuous time reparameterization, an interpretation is performed for the procedures of weak control variation in an auxiliary reduced problem, and new refinement methods are developed for impulsive processes. An example is provided.

We consider an optimal control problem for a hybrid dynamic system, where jumps of a trajectory may occur only at the moments of hitting a given closed subset of an extended phase space. A time reparameterization technique is developed to reduce the original problem to the one with bounded controls. We show that the reparameterized problem is equivalent to optimization in a class of generalized solutions to the hybrid system.

We consider an optimal control problem for an impulsive hybrid dynamic system, where jumps of a trajectory may occur only at the moments of hitting a given closed set. A time reparameterization technique is applied to reduce the original problem to the one with bounded controls. We show that the reparameterized problem is equivalent to optimization in a class of generalized solutions to the impulsive hybrid system. Necessary conditions for optimality of generalized control processes are obtained by interpreting the Maximum Principle in the reduced problem.

An optimal impulsive control problem is considered under constraint on the total impulse of control. In the problem there are conventional bounded controls and vector-valued measures (impulsive control). Based on a needle-shaped variation procedure an algorithm for optimal control of impulsive dynamic systems is elaborated. A discontinuous time reparameterization technique is applied. Control improvement property is established. Convergence analysis is provided.

We study the limit behavior of the reachable set for singularly perturbed nonautonomous linear systems with geometric control constraints. We assume that the system is stable in the fast variables and its coefficients are Lipschitz functions of time. We obtain estimates for the convergence rate as the small parameter tends to zero.

Optimal control problem for logic-dynamic systems is considered. Based on sufficient optimality conditions, similar to Krotov's conditions, a method for successive control improvement is developed. First- and second-order computational algorithms are constructed.

In this paper, asymptotic large-time dynamics of attainable sets to linear periodic control systems with bounded total impulse of control is studied. Asymptotic formulas for attainable sets and their shapes, i.e., the same sets considered up to an arbitrary nonsingular linear transformation, are found. It is shown that all limit shapes are obtained as the join of the limit shapes for neutral, stable-neutral, and unstable-neutral system components.

Explicit asymptotic formulas for reachable sets of linear dynamic systems with two types of control bounds, geometric constraints and constraints on the total impulse of the control action, are compared. A detailed proof of asymptotic formulas for the case of geometric constraints was first given. It was shown that, despite the similarity of the problem statement and methods to study the asymptotics of reachable sets for the considered types of constraints, the results obtained are essentially different. For example, for the impulsive case in the space of shapes of reachable sets, a multidimensional attractor may arise, while, in the case of geometric constraints, the corresponding attractor is reduced to a single point.

Explicit asymptotic formulas for the reachable sets of linear dynamical systems with constraints on the total impulse of control action under different assumptions concerning the spectrum of the system’s matrix are obtained. It is shown that for large time, the reachable sets can be approximately represented in the form of the product of the scaling matrix and the normalized reachable set, where the matrix is an elementary function of time, and the normalized reachable set depends on time quasi-periodically. Analysis of asymptotic formulas has shown that, generally speaking, there exists a continuum of limit shapes, the aggregate of which produces a multidimensional attractor.

An algorithm for polyhedral approximation of the reachable set of impulsive dynamic control systems is designed. The boundary points of the reachable set are determined by recursively generating and solving a family of auxiliary optimal impulsive control problems with state-linear objective functional. The impulsive control problem is solved with an algorithm that implicitly reduces the problem an ordinary optimal control problem. The reduced problem thus obtained is solved with an algorithm based on local approximations of the reachable set.

We study the long term asymptotic behavior of attainable sets and their shapes of linear time-invariant impulse control systems. We give an exhaustive description of attractors arising and the related dynamics. The results are compared with [4], [3].

The discrete problem of optimal control with mixed constraints was considered from the standpoint of numerical solution. Computational methods are usually based on solving a series of improvement problems. The nonlinear discrete problem and that of nonlinear functional improvement for the variational system were compared. A procedure for constructing an admissible control process improving the original problem from the solution of the reduced problem was described.

The terminal functional of a general control system is refined by studying an analogous problem for a variational system and regularization. A sequential refinement method is designed by combining the local approximation of the reachability set and reduction. The corresponding algorithm has relaxation properties. An illustrative example is given.

A study of continuous electrocardiographic and sphygmographic records in patients with atrial fibrillation demonstrated a regulatory effect of the heart's afferent systems. Computerized mathematical treatment of stroke output variation in pairs of adjacent shortened or lengthened (as compared to the mean) cycles showed the afferent systems to stabilize current minute volume while the body's circulatory requirements remained constant.