A simplification of the Agrachev–Gamkrelidze second-order variation for bang–bang controls

Ratmansky, Irad; Margaliot, Michael

doi:10.1016/j.sysconle.2009.11.002

Cited by 8 publications

(3 citation statements)

References 21 publications

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“…The calculation of the terms z 1 and z 2 in ( 27) requires two steps. The first is to derive an expression for the first-and second-order derivative of C(T ; s, α) with respect to s. This is based on the Agrachev-Gamkrelidze second-order variation for bang-bang controls [39] (see also [42], [43]). The second step is to derive an expression for the first-and second-order derivatives of the spectral radius of a matrix with respect to perturbations of the matrix entries.…”

Section: B Proof Of Theoremmentioning

confidence: 99%

High-order maximum principles for the stability analysis of positive bilinear control systems

Hochma

Margaliot

2015

Optim. Control Appl. Meth.

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We consider a continuous-time positive bilinear control system (PBCS), i.e. a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix C(t) is entrywise nonnegative for all time t ≥ 0. Motivated by the stability analysis of positive linear switched systems (PLSSs) under arbitrary switching laws, we fix a final time T > 0 and define a control as optimal if it maximizes the spectral radius of C(T ). A recent paper [2] developed a first-order necessary condition for optimality in the form of a maximum principle (MP). In this paper, we derive higher-order necessary conditions for optimality for both singular and bang-bang controls. Our approach is based on combining results on the secondorder derivative of the spectral radius of a nonnegative matrix with the generalized Legendre-Clebsch condition and the Agrachev-Gamkrelidze second-order optimality condition. Index TermsPositive switched systems, stability under arbitrary switching laws, variational approach, high-order maximum principles, Perron-Frobenius theory.

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Section: B Proof Of Theoremmentioning

confidence: 99%

High-order maximum principles for the stability analysis of positive bilinear control systems

Hochma

Margaliot

2015

Optim. Control Appl. Meth.

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“…These include the Pontrayagin maximum principle (PMP) and, in some cases, the Hamilton-Jacobi-Bellman equation [34,22]. The variational approach allows the application of sophisticated and powerful tools, such as first-and higher-orders maximum principles [1,12,42] to stability analysis. Some of the results can be generalized to nonlinear control systems and nonlinear switched systems.…”

Section: Problem 1 Find a Controlmentioning

confidence: 99%

“…. , u(0)} defined by u(q) = u * (q) for q odd, and u(l) = It is straightforward to verify that u steers the auxiliary system to the same final location as in (42) at time N − 1, i.e. y(N − 1; u) = y * (N ).…”

Section: Proposition 6 Suppose That For Anymentioning

confidence: 99%

Analysis of Discrete-Time Linear Switched Systems: A Variational Approach

Monovich¹,

Margaliot²

2011

SIAM J. Control Optim.

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A powerful approach for analyzing the stability of continuous-time switched systems is based on using tools from optimal control theory to characterize the "most unstable" switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific "most unstable" switching law. More generally, this so-called variational approach was successfully applied to derive nice-reachability-type results for both linear and nonlinear continuous-time switched systems.Motivated by this, we develop in this paper an analogous approach for discrete-time linear switched systems. We derive and prove a necessary condition for optimality of the "most unstable" switching law. This yields a type of discrete-time maximum principle (MP). We demonstrate using an example that this MP is in fact weaker than its continuous-time counterpart. To overcome this, we introduce the auxiliary system of a discrete-time linear switched system, and show that regularity properties of time-optimal controls for the auxiliary system imply nice-reachability results for the original discretetime linear switched system. Using this approach, we derive several new Lie-algebraic conditions guaranteeing nice-reachability results. These results, and their proofs, turn out to be quite different from their continuous-time counterparts.

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A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

Monovich

Margaliot

2011

Automatica

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A simplification of the Agrachev–Gamkrelidze second-order variation for bang–bang controls

Cited by 8 publications

References 21 publications

High-order maximum principles for the stability analysis of positive bilinear control systems

High-order maximum principles for the stability analysis of positive bilinear control systems

Analysis of Discrete-Time Linear Switched Systems: A Variational Approach

A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

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