Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subgradient evolution of value functions in discrete-time optimal control

Julio Deride, Cristopher Hermosilla, Mattia Solla

Abstract

In this paper we investigate how the subgradients of the value function of a discrete-time convex Bolza problem evolve over time. In particular, we develop a discrete-time version of the characteristic method introduced by Rockafellar and Wolenski in the 2000s, by showing that the time-evolution of the subgradients of the value functions can be associated with trajectories of a discrete-time Hamiltonian system. To do so, we first prove that the value function has a dual counterpart, which corresponds to the conjugate of the value function of a suitable dual problem. We finally make a discussion about the qualification conditions we require for our results, showing in particular that classical problems, such as the Liner-Quadratic regulator, satisfy these hypotheses.

Subgradient evolution of value functions in discrete-time optimal control

Abstract

In this paper we investigate how the subgradients of the value function of a discrete-time convex Bolza problem evolve over time. In particular, we develop a discrete-time version of the characteristic method introduced by Rockafellar and Wolenski in the 2000s, by showing that the time-evolution of the subgradients of the value functions can be associated with trajectories of a discrete-time Hamiltonian system. To do so, we first prove that the value function has a dual counterpart, which corresponds to the conjugate of the value function of a suitable dual problem. We finally make a discussion about the qualification conditions we require for our results, showing in particular that classical problems, such as the Liner-Quadratic regulator, satisfy these hypotheses.
Paper Structure (10 sections, 6 theorems, 74 equations)

This paper contains 10 sections, 6 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Notation and essentials
  3. Fully Convex Bolza problems
  4. Dual Bolza problem and weak duality
  5. Strong duality
  6. Discrete-time Characteristic method
  7. Examples
  8. Linear Quadratic problem with state constraints
  9. Linear-convex problem with mixed constraints
  10. Conclusion

Key Result

Proposition 2.2

For any $\tau\in[\![0:T]\!]$ we have that Moreover, we also have that $\omega_\tau\geq\vartheta_\tau^*$ and $\vartheta_\tau\geq\omega_\tau^*$.

Theorems & Definitions (13)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • Definition 3.1
  • Theorem 3.2
  • ...and 3 more