Asymptotic controllability and optimal control

Abstract

We consider a control problem where the state must approach asymptotically a target C while paying an integral cost with a non-negative Lagrangian l. The dynamics f is just continuous, and no assumptions are made on the zero level set of the Lagrangian l. Through an inequality involving a positive number View the MathML sourcep¯0 and a Minimum Restraint FunctionU=U(x)U=U(x) – a special type of Control Lyapunov Function – we provide a condition implying that (i) the system is asymptotically controllable, and (ii) the value function is bounded by View the MathML sourceU/p¯0. The result has significant consequences for the uniqueness issue of the corresponding Hamilton–Jacobi equation. Furthermore it may be regarded as a first step in the direction of a feedback construction

Stabilizability in optimal control

“…[22,25,33]). The main contribution of [19,23] was to prove that the existence of a p 0 -Minimum Restraint Function W for some p 0 > 0 allows to produce a pair (x, u) that meets both of these properties. A related and important goal is, given a p 0 -Minimum Restraint Function W for some p 0 > 0, to provide a state feedback K : R n \ C → U such that the systemẋ(t) = f (x(t), K(x(t)) is globally asymptotically stable to C and has (p 0 , W )-regulated cost, that is, such that for any stable trajectory x with x(0) = z, the cost Tx 0 l(x(t), K(x(t))) dt is not greater than W (z)/p 0 .…”

“…A related and important goal is, given a p 0 -Minimum Restraint Function W for some p 0 > 0, to provide a state feedback K : R n \ C → U such that the systemẋ(t) = f (x(t), K(x(t)) is globally asymptotically stable to C and has (p 0 , W )-regulated cost, that is, such that for any stable trajectory x with x(0) = z, the cost Tx 0 l(x(t), K(x(t))) dt is not greater than W (z)/p 0 . In this paper we address the question, left by [19,23] as an open problem, of how to define such a feedback law through the use of W . In the ideal case in which W is differentiable and there exists a continuous feedback K(x) such that DW (x) , f(x, K(x)) + p 0 l(x, K(x)) ≤ −γ(W (x)) for all x ∈ R n \ C, one easily derives global asymptotic stabilizability with (p 0 , W )-regulated cost.…”

Intelligent Information and Database Systems

“…The experiment involves the following two benchmark problems [32], which are defined on Table 1 cost of a product or process in a simpler, more efficient and systematic manner than traditional trial-and-error processes (Lin et al, 2009). Further, the statistical software MINITAB 14 was used in the analysis of parameter design for the SIG algorithm, where the signal-to-noise (S/N) ratio (Lin et al, 2009) is used to evaluate the stability of system quality in the experiment.…”

“…In the latter experiment, the SIG algorithm stops and the RBFN corresponding to the maximum fitness value are selected. After those critical parameter values (i.e., the number of hidden neuron, width, and weight) are set, RBFN can initiate the training and learning of approximation through two benchmark problems (i.e., B2 and Griewank continuous test functions [32]). …”

Solvability of some partial functional integrodifferential equations with finite delay and optimal controls in Banach spaces

“…Zhou ( 2014 ) considered a controlled stochastic delay partial differential equation with Neumann boundary conditions and studied the optimal control problem by means of the associated backward stochastic differential equations. In Motta and Rampazzo ( 2013 ), the authors discussed the assymptotic controllability and the optimal control of some control system where the state approaches asymptotically a target, while paying an integral cost with a nonnegative Lagrangian. Wang and Zhou ( 2011 ) discussed the optimal controls of a Lagrange problem for fractional evolution equations.…”