On control problems with restricted coordinates

Куржанский, А. Б.; Osipov, Iu.S.

doi:10.1016/0021-8928(70)90041-9

Cited by 9 publications

(6 citation statements)

References 2 publications

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“…It should also be noted that the scheme of solving linear problems of optimal control based on a moment problem makes it possible by means of the separability theorem of convex sets in function spaces to consider restrictions on the phase vector and the control (see, e.g., [156]). Solution of some linear problems of optimal control by reducing them by means of a moment problem to an equivalent problem of linear programming is given in the paper of Bondarenko and Filimonov [43].…”

Section: Computational Methods In Linear Problems Of Optimal Controlmentioning

confidence: 99%

Computational and approximate methods of optimal control

Chernousko¹,

Kolmanovskii²

1979

J Sov Math

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Section: Computational Methods In Linear Problems Of Optimal Controlmentioning

confidence: 99%

Computational and approximate methods of optimal control

Chernousko¹,

Kolmanovskii²

1979

J Sov Math

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“…The components u and θ of such a strategy are similar to the components u x and u t in the idealized control synthesis (13). The component u(t, x) specifies the direction of the control pulse produced on the interval [t, t + θ(t, x)].…”

Section: Definition 1 a Pairmentioning

confidence: 98%

“…The following assertions briefly outline the solution of the second subproblem in the form of a program control (for details, see [4,13]). Assertion 1.…”

Section: An Idealized Schemementioning

confidence: 99%

The Dynamic Programming Method in Impulsive Control Synthesis

2005

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The control synthesis problem is one of the central problems in modern control theory. Its solution can be obtained in various classes of feedback controls. For example, in classical control theory under a geometric constraint, the desired control ranges in the set of extreme points of the constraint, so that the synthesized system is described by differential equations with discontinuous right-hand side [1]. Sometimes, control synthesis can be described with the use of "switching surfaces" dividing the phase space into domains with continuous controls, while discontinuities ("switchings") are allowed on these surfaces [2][3][4].However, the solutions can have impulsive character in many applied problems, for example, in aerospace systems with instantaneous motion corrections, in systems with communication constrains, and in logical-dynamical systems. Such solutions require controls of generalized type, which consist of impulsive "delta functions" or their combinations with a bounded control. First program solutions in the impulsive control problem were obtained in [5]. It was shown in [6] that, in a linear impulsive problem, the number of jumps of an optimal control does not exceed the dimension of the phase space.Such problems were considered mainly from the viewpoint of program controls [4,7,8], while the construction of a well-formalized theory of impulsive control synthesis still remains an open problem. In the present paper, we show that dynamic programming methods can be applied to impulsive control problems in which solutions are sought in the form of synthesizing strategies. We consider linear systems, which permits one to combine the classical theory of distributions with the theory of generalized (viscosity) solutions [9-11] of the corresponding quasi-variational inequalities [12] of Hamilton-Jacobi-Bellman type. The suggested approach also permits one to study problems with higher-order derivatives of delta functions [13]. THE PROBLEMConsider the minimization problem for a generalized Mayer-Bolza functional on the trajectories of an impulsive control system:Here x(t) ∈ R n is the phase vector, U (·) ∈ BV ([t 0 , t 1 ] ; R m ) is a generalized control, and BV ([t 0 , t 1 ] ; R m ) is the space of functions of bounded variation ranging in R m . We assume that A(t) ∈ R n×n and B(t) ∈ R n×m are continuous matrix functions. The terminal time t 1 is fixed. The terminal term ϕ : R n → R ∪ {∞} is a closed convex function, whose presence in the expression for J(u(·)) permits us to state the optimality principle in what follows.The special choice 1 ϕ(x) = I (x| {x 1 }) of the function ϕ leads to the well-known problem of bringing a system from a point x 0 at time t 0 to a point x 1 at time t 1 with minimum variation of the control: Var [t0,t1] U (·) → inf, dx(t) = A(t)x(t)dt + B(t)dU (t), t∈ [t 0 , t 1 ] ,x (t 0 − 0) = x 0 , x(t 1 + 0) = x 1 .(2) 1 Here I (x|A) stands for the indicator function of a set A. (It vanishes on A and is equal to +∞ outside A.)

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“…This optimal control problem may be approached through the maximum principle [21,10,29]. To handle the state constraints X [τ ] one usually imposes the following constraint qualification.…”

Section: Problem 41 Calculate the Support Functionsmentioning

confidence: 99%

“…The "standard" maximum principle under state constraints[10,8].For Problem 4.1 under Assumption 4.1, the optimal control…”

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confidence: 99%

Ellipsoidal Techniques for Reachability Under State Constraints

Kurzhanski

Varaiya

2006

SIAM J. Control Optim.

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The paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with time-varying coefficients, bounds on the controls, and constraints on the state. The scheme provides tight external approximations by ellipsoidalvalued tubes. These tubes touch the reach tubes from the outside at each point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes. The result is an exact parametric representation of reach tubes through families of external ellipsoidal tubes. The parameters that characterize the approximating elliposids are solutions of ordinary differential equations with coefficients given partly in explicit analytical form and partly through the solution of a recursive optimization problem. The scheme combines the calculation of external approximations of infinite sums and intersections of ellipsoids, and suggests an approach to calculate reach sets of hybrid systems.

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On control problems with restricted coordinates

Cited by 9 publications

References 2 publications

Computational and approximate methods of optimal control

Computational and approximate methods of optimal control

The Dynamic Programming Method in Impulsive Control Synthesis

Ellipsoidal Techniques for Reachability Under State Constraints

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