High order Lyapunov-like functions for optimal control

Motta, Monica; Rampazzo, Franco

doi:10.1016/j.ifacol.2021.06.083

Cited by 3 publications

(4 citation statements)

References 15 publications

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“…3.6 also implies the result in [29], where a sketch is given of the fact that (under slightly stronger assumptions than those assumed here) the existence of a degree-k MRF yields GAC with regulated cost.…”

Section: Main Resultsupporting

confidence: 63%

“…From (20) it follows that for all q 1 , q 2 ∈ N, 1 ≤ q 1 < q 2 ≤ k, a degree-q 1 MRF (for some p 0 and γ) is also a degree-q 2 MRF (for the same p 0 and γ), while the converse is false, in general. In particular, a smooth degree-k MRF for some k > 1 may exist in situations where there are no smooth degree-1 MRFs (see the examples in [29,28]). Actually, this is one of the main reasons for considering degree-k MRFs with k > 1.…”

Section: Definition 33 (Degree-k Mrf)mentioning

confidence: 99%

“…A notion of (locally semiconcave) degree-k CLF was first introduced in [28]. The definition of degree-k MRF has been anticipated in [29], in the special case of constant p 0 and lagrangian l independent of the control a ∈ A.…”

Section: Definition 33 (Degree-k Mrf)mentioning

confidence: 99%

“…Let us observe that, as a trivial consequence of the monotonicity (1) , the higher the Hamiltonian's degree the larger the corresponding set of MRFs. Furthermore, as shown in an example in [29], for k sufficiently large it can well happen that a smooth degree-k MRF does exist while a standard, i.e. degree-1, C 1 MRF, does not (see also [28,Ex. 2.1-2.4], for the case with no cost).…”

Section: Introductionmentioning

confidence: 96%

See 3 more Smart Citations

HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation

Fusco¹,

Motta²,

Rampazzo³

2023

Preprint

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With reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree k ≥ 1, and we call any of its (suitably defined) solutions a degree-k Minimum Restraint Function. We prove that the existence of a degree-k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost.

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Section: Main Resultsupporting

confidence: 63%

Section: Definition 33 (Degree-k Mrf)mentioning

confidence: 99%

Section: Definition 33 (Degree-k Mrf)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 2 more Smart Citations

HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation

Fusco¹,

Motta²,

Rampazzo³

2023

Preprint

View full text Add to dashboard Cite

show abstract

HJ Inequalities Involving Lie Brackets and Feedback Stabilizability with Cost Regulation

2023

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With reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree $$k\ge 1$$ k ≥ 1 , and we call any of its (suitably defined) solutions a degree-kMinimum Restraint Function. We prove that the existence of a degree-k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost.

show abstract