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Finite horizon stochastic $H_2/H_\infty$ control for continuous-time mean-field systems with Poisson jumps

Huimin Han, Shaolin Ji, Weihai Zhang

TL;DR

This paper studies finite-horizon stochastic $H2/Hinf$ control for continuous-time mean-field jump-diffusion systems with Poisson jumps. It develops a mean-field bounded real lemma (MF-SJBRL) using quasi-linearization and completing-the-square to link feasibility to four cross-coupled generalized differential Riccati equations (GDREs). The main result yields time-varying linear feedback controls $u^*(t,x)$ and $v^*(t,x)$ with gains derived from the GDRE solutions, ensuring robust disturbance attenuation and optimal performance for interacting particle systems. A backward-in-time numerical scheme demonstrates the approach on a financial-portfolio example, validating solvability conditions and controller performance. The framework extends mean-field control with jumps and provides a systematic design tool for robust mean-field systems under stochastic disturbances.

Abstract

The stochastic $H_2/H_\infty$ control problem for continuous-time mean-field stochastic differential equations with Poisson jumps over finite horizon is investigated in this paper. Continuous and jump diffusion terms in the system depend not only on the state but also on the control input, external disturbance, and mean-field components. By employing the quasi-linear technique and the method of completing the square, a mean-field stochastic jump bounded real lemma of the system is derived, which plays a crucial role in solving stochastic $H_2/H_\infty$ control problem. It is demonstrated in this study that the feasibility of the stochastic $H_2/H_\infty$ control problem is equivalent to the solvability of four sets of cross-coupled generalized differential Riccati equations, thus generalizing the previous results to mean-field jump-diffusion systems. To validate the proposed methodology, a numerical simulation example is provided to illustrate the effectiveness of the control strategy. The results establish a systematic approach for designing $H_2/H_\infty$ controllers that simultaneously guarantee the robustness against disturbances and optimal performance for interacting particle systems.

Finite horizon stochastic $H_2/H_\infty$ control for continuous-time mean-field systems with Poisson jumps

TL;DR

This paper studies finite-horizon stochastic control for continuous-time mean-field jump-diffusion systems with Poisson jumps. It develops a mean-field bounded real lemma (MF-SJBRL) using quasi-linearization and completing-the-square to link feasibility to four cross-coupled generalized differential Riccati equations (GDREs). The main result yields time-varying linear feedback controls and with gains derived from the GDRE solutions, ensuring robust disturbance attenuation and optimal performance for interacting particle systems. A backward-in-time numerical scheme demonstrates the approach on a financial-portfolio example, validating solvability conditions and controller performance. The framework extends mean-field control with jumps and provides a systematic design tool for robust mean-field systems under stochastic disturbances.

Abstract

The stochastic control problem for continuous-time mean-field stochastic differential equations with Poisson jumps over finite horizon is investigated in this paper. Continuous and jump diffusion terms in the system depend not only on the state but also on the control input, external disturbance, and mean-field components. By employing the quasi-linear technique and the method of completing the square, a mean-field stochastic jump bounded real lemma of the system is derived, which plays a crucial role in solving stochastic control problem. It is demonstrated in this study that the feasibility of the stochastic control problem is equivalent to the solvability of four sets of cross-coupled generalized differential Riccati equations, thus generalizing the previous results to mean-field jump-diffusion systems. To validate the proposed methodology, a numerical simulation example is provided to illustrate the effectiveness of the control strategy. The results establish a systematic approach for designing controllers that simultaneously guarantee the robustness against disturbances and optimal performance for interacting particle systems.

Paper Structure

This paper contains 9 sections, 8 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Problem Formulation
  5. Three Useful Lemmas
  6. Mean-Field Stochastic Jump Bounded Real Lemma
  7. Stochastic $H_{2}/H_{\infty}$ control
  8. Numerical Simulation
  9. Conclusion

Key Result

Lemma 1

(see 1992 and 2019) Assume that $b:[0, T] \times \mathbb{R}^{n} \times \mathcal{P}_{2}(\mathbb{R}^{n})\rightarrow \mathbb{R}^{n}, \sigma:[0, T] \times \mathbb{R}^{n} \times \mathcal{P}_{2}(\mathbb{R}^{n}) \rightarrow \mathbb{R}^{n}$, $\pi:[0, T] \times G \times \mathbb{R}^{n} \times \mathcal{P}_{2}( where $\rho(\mu_{1},\mu_{2})$ is Wasserstein metric which satisfies Then the following equation wi

Figures (6)

  • Figure 1: The trajectories of $P_{1}$ and $Q_{1}$
  • Figure 2: The trajectories of $P_{2}$ and $Q_{2}$
  • Figure 3: The trajectories of $det(P_{1})$ and $det(Q_{1})$
  • Figure 4: The trajectories of $det(P_{2})$ and $det(Q_{2})$
  • Figure 5: The trajectories of $K_{1}$ and $K_{1}+\tilde{K}_{1}$
  • ...and 1 more figures

Theorems & Definitions (16)

  • Definition 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 1
  • ...and 6 more