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Singular extremals of optimal control problems with $L^1$ cost

Andrei Agrachev, Ivan Beschastnyi, Michele Motta

TL;DR

This work analyzes optimal control problems with an L1 cost for control-affine systems on smooth manifolds by applying Pontryagin's Maximum Principle to classify extremals into regular and singular types. It develops a geometric framework based on a super-Hamiltonian and a Jacobi equation to derive a strong generalized Legendre-Clebsch condition, proving that positivity of h_{c0c} together with absence of conjugate points yields local strong optimality for short singular arcs, and that the Legendre condition is necessary for optimality. The authors provide concrete geometric constructions, compute the Jacobi equation in several examples, and use the second variation to connect conjugate points with optimality via Morse theory in the singular setting. Collectively, the paper offers practical criteria for verifying local optimality of singular controls and demonstrates the theory through Heisenberg, Martinet, and SU(2) examples.

Abstract

We study the optimal control problem for a control-affine system, where we want to minimize the $L^1$ norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectories into two categories: regular and singular extremals. Then, we obtain a strong generalized Legendre-Clebsch condition for singular extremals and we show that this condition together with the absence of conjugate points is sufficient to ensure local strong optimality. We provide also some geometric examples where we apply our results. Finally, we prove that generalized Legendre-Clebsch condition is necessary for optimality.

Singular extremals of optimal control problems with $L^1$ cost

TL;DR

This work analyzes optimal control problems with an L1 cost for control-affine systems on smooth manifolds by applying Pontryagin's Maximum Principle to classify extremals into regular and singular types. It develops a geometric framework based on a super-Hamiltonian and a Jacobi equation to derive a strong generalized Legendre-Clebsch condition, proving that positivity of h_{c0c} together with absence of conjugate points yields local strong optimality for short singular arcs, and that the Legendre condition is necessary for optimality. The authors provide concrete geometric constructions, compute the Jacobi equation in several examples, and use the second variation to connect conjugate points with optimality via Morse theory in the singular setting. Collectively, the paper offers practical criteria for verifying local optimality of singular controls and demonstrates the theory through Heisenberg, Martinet, and SU(2) examples.

Abstract

We study the optimal control problem for a control-affine system, where we want to minimize the norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectories into two categories: regular and singular extremals. Then, we obtain a strong generalized Legendre-Clebsch condition for singular extremals and we show that this condition together with the absence of conjugate points is sufficient to ensure local strong optimality. We provide also some geometric examples where we apply our results. Finally, we prove that generalized Legendre-Clebsch condition is necessary for optimality.

Paper Structure

This paper contains 19 sections, 17 theorems, 236 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Description of extremal trajectories
  3. Definitions and notations
  4. Description of extremal trajectories
  5. The super-Hamiltonian function and sufficient conditions for strong local optimality of short arcs
  6. Jacobi equation and sufficient conditions for optimality
  7. Linear symplectic geometry and conjugate points
  8. Sufficient condition for optimality
  9. Examples
  10. An example: the Heisenberg group with a drift
  11. A Martinet type example
  12. Left-invariant problem in SU(2)
  13. Second variation and the generalized Legendre condition
  14. Definition of the end-point map and the Hessian
  15. Explicit expression for the second variation form
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $u\in\mathcal{U}$ be a singular control which is also an optimal solution of Problem prob:OCP. Let $\lambda(\cdot)$ be its corresponding extremal solving the Hamiltonian system eq:HamSystPMP of the PMP. Define $h_c=\sum_{i=1} ^m h_i^2$. Then

Figures (3)

  • Figure 1: Graphical representation of Local Strong Optimality
  • Figure 2: Construction of $h_S$.
  • Figure 3: Graphical explanation on how the initial subspace $L_0$ should be chosen, for $n=1$ (so, $L(T_{\lambda_0}(T^*M)) \simeq S^1$). Supposing that a curve with positive velocity moves clockwise, you see that if we choose $L_0$ in the right-half of $\mathcal{V}$ (case (a)), then $B_t L_0$ does not cross the vertical space $\Pi_0$ for small times. If instead we choose $L_0$ in the left-half of $\mathcal{V}$, then there can be an intersection with $\Pi_0$ in arbitrarily small time.

Theorems & Definitions (32)

  • Theorem 1.1: Necessary optimality condition
  • Definition 1.2: Local strong optimality
  • Theorem 1.3: Sufficient conditions, small time
  • Definition 1.4: Corank of an extremal
  • Theorem 1.5: Necessary condition, conjugate points
  • Theorem 1.6: Sufficient condition, absence of conjugate points
  • Theorem 2.1: Pontryagin Maximum Principle
  • Theorem 3.1: Stefani, Zezza
  • proof : Proof of Theorem \ref{['thm:suff-cond-small-time']}
  • Definition 4.1
  • ...and 22 more