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Two-Player Zero-Sum Hybrid Games with Set-Valued Dynamics

Santiago J. Leudo, Francesco Ferrante, Ricardo G. Sanfelice

Nov 3, 2026arXiv:2611.04999v1
math.OCeess.SY
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Scorecard· 16/16
4.5/10 impact

Competent, rigorous but incremental extension of the authors' own hybrid-game framework to set-valued dynamics, with sufficiency-only results and limited validation.

Abstract

In this paper, we formulate a two-player zero-sum game under dynamic constraints defined in terms of a hybrid inclusion. The game consists of a min-max problem involving a cost functional that depends on the actions and corresponding (potentially non-unique) solutions, defined as functions of hybrid time and, hence, can flow or jump. We present sufficient conditions given in terms of Hamilton-Jacobi-Isaacs-like equations to establish a bound and to exactly evaluate the worst-case cost under the optimal strategy. Under additional conditions, we show that the proposed optimal state-feedback laws render a set of interest pre-asymptotically stable for the resulting hybrid closed-loop system. The results are illustrated in an application with sporadic jumps and linear quadratic hybrid dynamics.

AI Impact Assessments

(1 model)

Core Contribution

The paper extends two-player zero-sum hybrid games (Leudo & Sanfelice, Automatica 2026) to hybrid inclusions, where flow and jump maps are set-valued and solutions are not unique. Nonuniqueness makes the cost of a given input ill-defined, so the authors define the cost as the supremum over maximal solutions (the worst case). They then introduce a "weak" feedback saddle-point equilibrium. Their results are:

  • Lyapunov/HJI-like sufficient conditions that upper-bound the cost (Prop. 4.2, 5.1).
  • Exact cost evaluation via an auxiliary "maximal" hybrid system whose solutions select the worst-case flow and jump (Cor. 4.3, Prop. 5.2).
  • Theorem 5.1: a function V satisfying HJI-like equalities and inequalities for flows and jumps, together with a terminal-limit condition, yields a weak saddle point and J* = V.
  • Theorem 6.1: if V is also sandwiched by class-K∞ functions and the stage costs are positive definite, the value function acts as a Lyapunov function and the saddle strategy renders a closed set pre-asymptotically stable.
  • A hybrid Riccati equation for linear-quadratic problems with timer-driven spontaneous jumps, and a scalar set-valued example.

    • The main novelty is the relaxation of solution uniqueness and the "weak" equilibrium notion. This is a modest, natural generalization of the authors' own previous work and of the Bernstein/Ferrante–Sanfelice cost-evaluation line.

    Methodological Rigor

    The proofs follow the standard telescoping Lyapunov argument along hybrid time domains, and the logic is mostly sound for sufficiency results. There are some concerns:

  • The proof of Theorem 5.1 is convoluted. Items b–d rely on infima over restricted strategy sets (S^s, S^w), and the text itself notes that V is not guaranteed to lower-bound every solution under u^s.
  • The resulting equilibrium is "weak": the worst-case cost is not necessarily attained, and the paper concedes that κ is not necessarily optimal in the min-max sense. Example 7.1 shows other gains giving lower cost when P2 sticks to κ2. This weakens the claim that the equilibrium is meaningful.
  • The Hamiltonian conditions use a Clarke-gradient-based Ṽ, and the maximal system requires attainment and existence results that are only sketched via references.
  • Some notational sloppiness: Corollary 7.1 states R_C1 and −R_C2 positive definite with a mismatched dimension index, and "V∘ϕ bounded" appears where V°ϕ was likely intended.
  • Several conditions are hard to verify in practice because V must be found, which the authors acknowledge.

    • The numerical evidence is a single scalar example plus a Riccati corollary. Corollary 7.1 itself contains no numerical test, and the linear-quadratic case is single-valued. The set-valued example is simple enough to be solved nearly by hand.

    Potential Impact

    The audience is narrow: the hybrid-systems control community (Sanfelice, Goebel, Teel framework), with some reach into robust and H∞-type control, security, and cyber-physical systems with intermittent communication. The results are sufficient conditions requiring a known value function, so applicability depends on constructing V. This is feasible mainly for linear-quadratic structure, where the hybrid Riccati result is useful as a design tool for robust control with random-interval jumps. Set-valued dynamics widen the model class (uncertainty, nondeterministic resets, differential inclusions). No computational method is provided, such as numerical HJI solvers or learning approaches.

    Timeliness & Relevance

    Games on hybrid dynamics are of moderate interest, given adversarial CPS security and robust control, but the paper is an incremental journal consolidation of conference papers (CDC 2020, ACC 2024) with complete proofs. It does not address a major current bottleneck.

    Strengths

  • It gives a coherent, rigorous formal framework consistent with the established hybrid-systems formalism.
  • It handles nonunique solutions explicitly and honestly, including a candid discussion of weak optimality.
  • It connects optimality to stability, using the value function as a Lyapunov function.
  • It includes a hybrid Riccati equation for a practically motivated class.
  • The appendix gives full proofs and existence conditions.

    • Limitations

  • Results are sufficiency-only, with no existence theory for V or the equilibrium.
  • The weak equilibrium concept is arguably unsatisfying, since the saddle point is not necessarily optimal.
  • Verification is hard, with little guidance on constructing V beyond LQ.
  • Numerical validation is thin and there is no comparison to alternatives.
  • There is no scalability discussion.
  • It is incremental relative to the authors' prior Automatica paper, with the main change being relaxation of uniqueness.
  • Some typographical and notational errors, and the exposition is dense.

    • Other Observations

    Reproducibility is moderate: the scalar example gives its parameters, and the Riccati equation is explicit. No code is mentioned. The paper is mathematically standard for the hybrid-systems community, with moderate technical difficulty. There is no refutation of prior claims. It supersedes earlier conference versions rather than independently replicating anything.

    Rating:4.5/ 10
    Significance 4Rigor 5.5Novelty 4.5Clarity 5

    Generated Nov 4, 2026

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