A Path-Complete Approach for Optimal Control of Switched Systems

Abstract: We study the problem of estimating the value function of discrete-time switched systems under arbitrary switching. Unlike the switched LQR problem, where both inputs and mode sequences are optimized, we consider the case where switching is exogenous. For such systems, the number of possible mode sequences grows exponentially with time, making the exact computation of the value function intractable. This motivates the development of tractable bounds that approximate it. We propose a novel framework, based on path-complete graphs, for constructing computable upper bounds on the value function. In this framework, multiple quadratic functions are combined through a directed graph that encodes dynamic programming inequalities, yielding convex and sound formulations. For example, for switched linear systems with quadratic cost, we derive tractable LMI-based formulations and provide computational complexity bounds. We further establish approximation guarantees for the upper bounds and show asymptotic non-conservativeness using concepts from graph theory. Finally, we extend the approach to controller synthesis for systems with affine control inputs and demonstrate its effectiveness on numerical examples.

• The paper introduces path-complete graph certificates that convert worst-case dynamic programming inequalities into tractable upper bounds for arbitrarily switched systems.
• The method yields polynomial-time semidefinite programs, and dual De Bruijn graphs provide asymptotically exact bounds with empirical relative-error factors improving to roughly 1.0007–1.009 at order four.
• The framework extends to controlled switched systems by synthesizing piecewise-linear policies with certified performance, while leaving controlled-case accuracy guarantees and broader templates as open problems.

This paper develops a path-complete framework for computing tractable upper bounds on the value function of discrete-time switched systems under arbitrary switching (2602.04310). The setting is distinct from the well-studied switched LQR literature: there, both the input and the mode sequence are decision variables, whereas here the switching signal is exogenous and adversarial. The value function is defined as the worst-case total cost over all mode sequences, and its computation is intractable because the number of candidate sequences grows exponentially in time; moreover, even for quadratic costs the value function is generally neither quadratic nor smooth, so no Riccati-type closed form exists.

Dynamic programming inequalities for nondeterministic systems

The authors first derive Bellman-type characterizations adapted to the worst-case (nondeterministic, nonstochastic) setting. The exact value function satisfies $J(x) = c(x) + \max_i J(f_i(x))$, and two relaxations of this equation yield one-sided certificates. Any function $V$ satisfying the inequality $V(x) \ge c(x) + \max_i V(f_i(x))$ globally upper-bounds $J$ by a telescoping argument along arbitrary trajectories. Conversely, a continuous function $W$ with $W(0)=0$ satisfying the reverse inequality lower-bounds $J$, provided all system trajectories converge to the origin. These two propositions are the workhorses of the paper: the first supplies sound upper bounds, the second underpins the approximation guarantees obtained later.

Path-complete upper bounds

The central construction transplants the path-complete graph framework of Ahmadi et al., originally devised for stability analysis via multiple Lyapunov functions, to value-function bounding. Each node $\alpha$ of a directed labeled graph carries a function $V_\alpha$ from a template (e.g., quadratics), and each edge $(\alpha,\beta,i)$ encodes the dynamic programming inequality $V$0. If the graph is complete, the pointwise minimum of the node functions satisfies the global Bellman upper inequality; if it is co-complete, the pointwise maximum does so. This gives a family of certified upper bounds parameterized by the graph, whose size can be increased to trade computational cost against conservativeness — a flexibility absent from Rantzer's earlier scheme, where quadratic functions are attached to modes and the inequality structure is fixed.

LMI formulations and complexity

Specializing to autonomous switched linear systems with quadratic cost $V$1 and a quadratic template, the edge inequalities reduce to the LMIs $V$2. The resulting semidefinite programs are solvable in polynomial time; with a trace objective, the paper reports an explicit complexity bound polynomial in $V$3, $V$4, and $V$5. Objective choice matters: minimizing the bound at a specific $V$6 yields a nonconvex program that can nevertheless be convexified by splitting into subproblems, while global objectives such as trace or log-det penalize large matrices across the state space.

Asymptotic non-conservativeness and approximation guarantees

Two results address tightness. First, using dual De Bruijn graphs $V$7 of growing order, the authors prove convergence: assuming asymptotic stability, there exist feasible quadratic solutions on these co-complete graphs such that $V$8 converges pointwise to the true value function as $V$9. The proof constructs feasible pieces from truncated horizon sums scaled by a factor $V(x) \ge c(x) + \max_i V(f_i(x))$0, where $V(x) \ge c(x) + \max_i V(f_i(x))$1 decays geometrically with the stability rate. Notably, this guarantees only existence of non-conservative feasible solutions; the solution returned by a particular optimization program may remain conservative depending on its objective. Second, the paper provides finite-order relative accuracy certification: solving additional SDPs based on S-procedure multipliers yields a factor $V(x) \ge c(x) + \max_i V(f_i(x))$2 with $V(x) \ge c(x) + \max_i V(f_i(x))$3 for all $V(x) \ge c(x) + \max_i V(f_i(x))$4. The min case requires more multipliers than the max case but follows the same pattern. Numerically, averaged over 500 random realizations, $V(x) \ge c(x) + \max_i V(f_i(x))$5 falls from roughly 1.04–1.30 at order 1 to 1.0007–1.009 at order 4 across configurations up to $V(x) \ge c(x) + \max_i V(f_i(x))$6, $V(x) \ge c(x) + \max_i V(f_i(x))$7 — strong empirical evidence of rapid tightening.

Extension to controller synthesis

For controlled systems $V(x) \ge c(x) + \max_i V(f_i(x))$8 with affine inputs, the framework directly synthesizes a piecewise linear policy $V(x) \ge c(x) + \max_i V(f_i(x))$9, where the active gain minimizes $J$0 over nodes. If the node functions and gains satisfy graph-driven inequalities on a complete graph, the resulting pointwise minimum certifies simultaneously an upper bound on the closed-loop value function $J$1 and hence on the optimal value function $J$2. The matrix inequalities become nonlinear through products involving gains but are converted to LMIs via the change of variables $J$3, $J$4 together with Schur complements. Experiments on a benchmark two-mode system show monotone decrease of the bound at two initial states as the De Bruijn order grows from 1 to 5 (e.g., from 9.428 to 8.813 under the pointwise objective), though monotonicity is provable only for the pointwise objective and not for log-det.

Limitations and open questions

The paper concedes several restrictions. The convergence theorem requires asymptotic stability and establishes only feasibility of a converging sequence, not that the optimizer of any given SDP converges. Approximation guarantees ($J$5-scaling lower bounds) are developed only for the autonomous case; whether analogous certificates exist for the controlled setting is left open. In the controlled case, convexification of the trace objective is unresolved, and the provable monotonicity of bounds with graph order covers only the pointwise objective. The extension beyond quadratic templates and beyond De Bruijn graph families for guaranteed convergence rates also remains unaddressed.

Conclusion

The paper provides a principled bridge between path-complete Lyapunov analysis and worst-case cost-to-go estimation: dynamic programming inequalities encoded on path-complete graphs yield convex, soundly composable upper bounds on the value function of arbitrarily switched systems, with polynomial-time LMI implementations, finite-sample accuracy certificates, and proven asymptotic exactness on dual De Bruijn graphs. The controlled-case extension delivers stabilizing piecewise-linear policies with certified performance bounds, positioning the framework as a tractable alternative to exponentially costly enumeration of switching sequences.