On the optimal second order decrease rate for nonlinear and symmetric control systems

Mauro Costantini; Pierpaolo Soravia; Mauro Costantini; Pierpaolo Soravia

doi:10.3934/math.20241369

AIMS Mathematics

2024, Volume 9, Issue 10: 28232-28255. doi: 10.3934/math.20241369

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On the optimal second order decrease rate for nonlinear and symmetric control systems

Mauro Costantini ,
Pierpaolo Soravia ^,

Dipartimento di Matematica, Università di Padova, via Trieste 63, Padova 35121, Italy

Received: 12 August 2024 Revised: 14 September 2024 Accepted: 14 September 2024 Published: 29 September 2024
MSC : Primary 93B05; Secondary 35F21, 13P25, 34L15

When a control system has all its vector fields tangent to the level set of a given smooth function $ u $ at a point $ \hat x $, under appropriate assumptions that function can still have a negative rate of decrease with respect to the trajectories of the control system in an appropriate sense. In the case when the system is symmetric and $ u $ has a decrease rate of the second order, we characterise this fact and investigate the existence of a best possible rate in the class of piecewise constant controls. The problem turns out to be purely algebraic and depends on the eigenvalues of matrices constructed from a basis matrix whose elements are the second order Lie derivatives of $ u $ at $ \hat x $ with respect to the vector fields of the system.
Citation: Mauro Costantini, Pierpaolo Soravia. On the optimal second order decrease rate for nonlinear and symmetric control systems[J]. AIMS Mathematics, 2024, 9(10): 28232-28255. doi: 10.3934/math.20241369

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Abstract

When a control system has all its vector fields tangent to the level set of a given smooth function $ u $ at a point $ \hat x $, under appropriate assumptions that function can still have a negative rate of decrease with respect to the trajectories of the control system in an appropriate sense. In the case when the system is symmetric and $ u $ has a decrease rate of the second order, we characterise this fact and investigate the existence of a best possible rate in the class of piecewise constant controls. The problem turns out to be purely algebraic and depends on the eigenvalues of matrices constructed from a basis matrix whose elements are the second order Lie derivatives of $ u $ at $ \hat x $ with respect to the vector fields of the system.

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