Minimax Optimal Control

Updated 12 December 2025

Minimax optimal control problems are robust formulations that minimize worst-case cost by counteracting adversarial disturbances.
The method employs dynamic programming and a linear Bellman equation to derive explicit feedback controllers under box constraints.
Scalable computational algorithms enable decentralized control design for large-scale networks and infrastructure systems.

A minimax optimal control problem is a class of optimal control problem in which the controller seeks to minimize a cost functional under the worst-case realization of an adversarial disturbance or model uncertainty. In this setup, "minimax" refers to the two-player, zero-sum game structure: the controller (minimizer) selects a control law to minimize the worst-case (maximized) cost that could be induced by allowable disturbances or model permutations. The minimax paradigm is particularly central in robust control, adversarial machine learning, and applications requiring guarantees against uncertainty or perturbations.

1. Fundamental Formulation and Structural Assumptions

A typical minimax optimal control problem is formulated as follows:

State evolution: Discrete or continuous-time dynamical system, often linear, e.g., $x(t+1) = A x(t) + B u(t) + F w(t)$ for discrete time, where $x$ is the state, $u$ the control input, and $w$ the disturbance.
Cost functional: The objective is to minimize the supremum over allowable disturbances $w$ :

$J_\mu(x_0) = \sup_{w(\cdot) \in \mathcal{W}} \sum_{t=0}^{\infty} \left[ s^\top x(t) + r^\top u(t) - \gamma^\top w(t) \right]$

Constraints: Control and disturbance policies are constrained element-wise by state-dependent "boxes," e.g., $|u(t)| \leq E x(t)$ , $|w(t)| \leq G x(t)$ , and $(x, u, w)$ are restricted to the positive orthant.

Key structural assumptions include:

Positivity: State and control variables are required to be nonnegative; system matrices $B, F, E, G$ are nonnegative.
Homogeneous, monotone constraints and costs: The cost and constraints are positively homogeneous and monotone in the state.
Stabilizability of Dynamics: The system matrix $A$ satisfies a dominance condition, e.g., $A \geq |B|E + |F|G$ elementwise, ensuring invariance of the nonnegative orthant.

These assumptions enable the explicit recognition of minimax optimal control problems as dynamic games structured for tractable analysis (Gurpegui et al., 2023, Gurpegui et al., 3 Feb 2025, Gurpegui et al., 7 Nov 2024).

2. Dynamic Programming Characterization and the Bellman Equation

The minimax problem admits a dynamic programming (DP) formulation. The value function is

$J^*(x) = \inf_{\mu} \sup_{w} \sum_{t=0}^{\infty} g(x(t), u(t), w(t))$

with $g(x, u, w)$ the stage cost.

The Bellman equation takes the two-player (minimax) form: $J^*(x) = \min_{|u|\leq E x} \max_{|w|\leq G x} \left[ s^\top x + r^\top u - \gamma^\top w + J^*(A x + B u + F w) \right]$

A central result in recent literature is that, under the aforementioned positivity and monotonicity assumptions (and appropriate cost-boundedness), the value function is exactly linear: $J^*(x)=p^\top x$ , $p \in \mathbb{R}_+^n$ (Gurpegui et al., 2023, Gurpegui et al., 3 Feb 2025, Gurpegui et al., 7 Nov 2024). This reduces the functional Bellman equation to a finite-dimensional vector fixed-point equation.

Substituting the linear ansatz yields: $p = s + A^\top p - E^\top | r + B^\top p | + G^\top | -\gamma + F^\top p |$ Solvability typically requires an additional feasibility condition ensuring the disturbance penalty is sufficiently strong: $\gamma \geq F^\top p$ (Gurpegui et al., 3 Feb 2025).

3. Explicit Minimax Feedback Controller Synthesis

Once $p$ is computed, the minimizer's optimal control law is given explicitly. For linear box constraints $|u| \leq E x$ , the Bellman minimization in each $u_i$ is separable with solution: $u^*_i = -\operatorname{sign}(r_i + p^\top B_i) E_i x$ The corresponding gain matrix $K$ is

$K = \begin{bmatrix} \operatorname{sign}(r_1 + B_1^\top p) E_1 \ \vdots \ \operatorname{sign}(r_m + B_m^\top p) E_m \end{bmatrix}$

with $u^* = -K x$ .

Notably, the structure and sparsity of $K$ directly inherit from $E$ —enabling structurally constrained or decentralized controller design for large-scale networked systems (Gurpegui et al., 2023, Gurpegui et al., 7 Nov 2024). The controller (as well as the value function) remains piecewise linear even for multi-disturbance formulations or continuous-time analogs (Gurpegui et al., 7 Nov 2024).

4. Computational Methods: Value Iteration and Scalability

The vector fixed-point equation for $p$ is solved via a monotone value-iteration algorithm:

p = 0
while True:
    q = r + B.T @ p
    z = -γ + F.T @ p
    p_new = s + A.T @ p - E.T @ abs(q) + G.T @ abs(z)
    if np.max(abs(p_new - p)) <= ε:
        break
    p = p_new

Each iteration is computationally light: a few matrix-vector multiplications that fully exploit sparseness in

A, B, F, E, G

(Gurpegui et al., 2023). Convergence is monotonic, and for well-conditioned systems, the number of iterations (dictated by the spectral radius of the iteration map) remains moderate even for high-dimensional networks.

This computational tractability extends to continuous-time systems via discretization and to LP-based formulations for the infinite-horizon linear regulator case (Gurpegui et al., 7 Nov 2024).

5. Problem Extensions, Robustness, and Primal-Dual Structure

Minimax optimal control theory supports several extensions:

Multiobjective and vector-valued objectives: Via the vector minimax DP and Blackwell approachability, robust control for vector cost criteria is realized with guarantees against adversarial model strategies (Kamal, 2010).
Distributionally robust and stochastic variants: When cost and dynamics contain uncertain or ambiguous parameters, minimax criteria are implemented via duality and moment-based ambiguity sets (Ye et al., 2016).
Nonlinear, ensemble, and symbolic settings: With abstraction, minimax solutions are computable for nonlinear or infinite-ensemble control systems via principles like $\Gamma$ -convergence and set-valued Bellman equations (Reissig et al., 2017, Scagliotti, 9 May 2024).
Reduction to minimization problem: When the disturbance penalty is sufficiently high (i.e., $\gamma \geq F^\top p$ ), the adversarial effect vanishes and the problem effectively collapses to the minimization (deterministic) optimal control counterpart (Gurpegui et al., 3 Feb 2025, Gurpegui et al., 7 Nov 2024).

The duality between disturbance robustness and constraint structure is reflected throughout these results, with LP or Riccati frameworks depending on system structure.

6. Application Example and Scalability

The minimax optimal control of positive systems is concretely illustrated by the voltage (DC) control of an electric power network (Gurpegui et al., 2023):

Discretization of networked differential equations yields state-space models with positive, sparse matrices due to physical constraints (e.g., conservation laws, graph connectivity).
Minimax controllers computed via value iteration achieve decentralized feedback gains that respect the sparsity and locality of network physical interconnections.
Empirical simulation shows minimax policy performance matches brute-force adversarial simulations; cost bounds are tight and scale gracefully with system size (Gurpegui et al., 2023, Gurpegui et al., 7 Nov 2024).

This framework extends to large-scale water network control, where minimax controllers yield scalable, robust stabilization with computational effort proportional to problem sparsity and iteration count (Gurpegui et al., 7 Nov 2024).

7. Theoretical and Practical Significance

Minimax optimal control problems, particularly for positive and monotone systems, advance robust control theory by:

Enabling explicit, closed-form solutions to dynamic-programming equations, contrasting with the generic intractability of general nonlinear minimax DP.
Delivering controllers with structural constraints directly embedded, thus supporting scalable computation and decentralized implementation.
Providing necessary and sufficient scalings of disturbance penalties ( $\gamma \geq F^\top p$ ) to guarantee finiteness and problem well-posedness.
Admitting value-iteration/LP-based computation in high dimensions, with theoretical guarantees for convergence and scalability.

Recent results bridge minimax optimal control to convex optimization, game theory, distributional robustness, and applications ranging from infrastructure to adversarial machine learning, establishing the minimax paradigm as a cornerstone of modern robust and network control theory (Gurpegui et al., 2023, Gurpegui et al., 3 Feb 2025, Gurpegui et al., 7 Nov 2024, Kamal, 2010).