Strictly Convex Control Sets

• Strictly convex control sets are compact convex subsets whose boundaries lack nontrivial line segments, ensuring each boundary point has a unique supporting hyperplane.
• In control system design, these sets guarantee unique optimal controls and improve stability by simplifying both theoretical and numerical analyses.
• Their well-behaved geometric and analytic properties facilitate efficient computation and robust optimization in decentralized, nonlinear, and hybrid control applications.

A strictly convex control set is a compact, convex subset of a real vector space with the property that its boundary contains no nontrivial line segments—that is, every boundary point has a unique supporting hyperplane or, equivalently, every supporting set in any given direction is a singleton. In control theory and related fields, imposing strict convexity on control sets ensures the uniqueness of optimal controls, enhances stability, regularizes geometric and analytic properties of reachable or invariant sets, and can simplify numerical and theoretical analysis. The prominence of strictly convex control sets spans nonlinear optimal control, convex analysis, geometric control, decentralized controller synthesis, computational geometry, and convex inner approximations for controller design.

1. Formal Definitions and Characterizations

Strict convexity can be characterized both geometrically and analytically. Given a real topological vector space $V$ and a convex compact set $U \subset V$, strict convexity requires that for any two distinct $x, y \in \partial U$ (the boundary), the open segment between $x$ and $y$ lies in the interior: $(1-t)x + t y \in \mathrm{int}(U)$ for all $t \in (0,1)$. This excludes any flat segments on $\partial U$.

When $U$ is the unit sublevel set of a Minkowski norm $N$ (i.e., $U = \{x : N(x) \leq 1\}$), strict convexity of $N$ is equivalent to the geometric condition above and also to the analytic property that for any $\alpha > 1$, the power $N^\alpha$ is strictly convex: $$N((1-t)x + t y)^\alpha < (1-t) N(x)^\alpha + t N(y)^\alpha \quad \forall\, x \neq y,\, t\in (0,1).$$ This ensures that average controls are strictly "better" (i.e., have strictly lower cost, in the case $N$ is a cost function) than their endpoints, forbidding multi-valuedness in optimal control. These characterizations are established in (Simon et al., 2022).

2. Metric Structure, Regularity, and Parametric Control Sets

The structure of the set of strictly convex control sets is particularly well-behaved under suitable metrics. The Pliś (or Demyanov) metric is defined for convex compacta $A, B \subset \mathbb{R}^n$ via

$p(A, B) = \sup_{\|p\|=1} h(A(p), B(p))$

where $A(p)$ denotes the face of $A$ maximizing $\langle p, \cdot \rangle$ and $h(\cdot, \cdot)$ is the Hausdorff distance. For strictly convex compacta, all supporting faces reduce to singletons, and the Pliś metric renders the set of strictly convex compact sets complete (Balashov et al., 2011).

This has consequences for parametric families of control sets, set-valued analysis, and robust optimization: convergence in the Pliś metric preserves strict convexity, Lipschitz selectors (e.g., the Steiner point) enjoy continuous (even Hölder) regularity under this metric, and the addition or linear transformation of strictly convex control sets is well-behaved, ensuring stability of geometric and control-theoretic properties under standard set operations.

3. Strict Convexity in Control System Design

3.1 Synthesis and Optimization

Many modern controller synthesis and control allocation problems are formulated with constraints described by strictly convex control sets. For general nonlinear and uncertain systems, if the admissible control set at each time is strictly convex and the Hamiltonian is strictly convex in the control variable, then necessary optimality conditions (Pontryagin's Maximum Principle) yield a unique minimizing control at each instant, ensuring existence and uniqueness (when further smoothness conditions hold) of optimal solutions (Abhijeet et al., 12 Apr 2024, Rossa et al., 15 Oct 2025). In semialgebraic controller design, strictly convex inner approximations are produced by trimming away concave regions from feasibility sets via a second-order boundary curvature analysis, allowing robust synthesis through convex programming (Henrion et al., 2011).

3.2 Decentralized and Structured Control

For structured or decentralized controller design, strictly convex control sets are central to guaranteeing tractability via the concept of quadratic invariance. The set of achievable closed-loop maps or Youla parameters is convex if and only if the allowable controller class is quadratically invariant under the plant (Lessard et al., 2013). This strong convexity is essential for practical synthesis via convex optimization.

3.3 Hybrid and Nonlinear Control

For hybrid or switched systems and algebraic or polynomial systems, the invariance (or viability) of a set is naturally reformulated geometrically via support functions. For strictly convex sets, the support function is differentiable away from zero, and the dual invariance conditions (such as

$$\min_{u \in U} \nabla_y p_C(y)^\top (A \nabla_y p_C(y) + B u) \leq 0, \;\forall y\neq 0$$

with $p_C$ the support function of $C$) provide less conservative and more computationally tractable certificates compared to classical sum-of-squares or LMI approaches (Legat et al., 2021, Legat et al., 2021).

4. Computation and Geometry of Strictly Convex Control Sets

In computational geometry, the structure and algorithms for ball hulls and ball intersections in strictly convex normed planes—even extending to general Minkowski spaces—are relevant for optimal coverage, localization, and sensor placement problems. The strict convexity ensures uniqueness of minimal arcs (no flat sides) between boundary points, guaranteeing that generic computational geometry algorithms for centers, hulls, and intersections (e.g., O($n \log n$) for construction) remain efficient (Martín et al., 2014, Martín et al., 2014).

5. Analytical and Topological Properties

Strictly convex control sets admit strictly convex, lower semicontinuous functions, yielding rich analytic structure: every compact strictly convex set embeds in a strictly convex dual Banach space with weak* topology, and there is a dense set of exposed points (extreme points with unique supporting functionals) where the strictly convex function is continuous (García-Lirola et al., 2015). This has implications for uniqueness and stability in optimization and variational analysis.

In finite-dimensional settings, the geometric regularity of strictly convex control sets enables rectifiability of regular subsets: the density equals 1 almost everywhere implies $m$-rectifiability, so sets with full measure-theoretic density can locally be approximated by $m$-planes—fundamental for stability, controllability, and reachability analysis via geometric measure theory (Wilson, 2023).

6. Geometric Control and Zermelo Navigation: Nonparametric and Asymmetric Models

Allowing the admissible velocity set $U$ in a navigation problem to be an arbitrary strictly convex compact set significantly generalizes classical models. Existence of optimal solutions is assured under appropriate weak-current conditions. The strict convexity ensures that the control maximizing the Pontryagin Hamiltonian is unique and smooth as a function of the costate, enabling elimination of dual variables (in 2D) and yielding second-order differential equations for the velocity vector field (e.g., $\langle u'^\perp, u + s(x)\rangle = 0$) that generalize the classical Zermelo equation. In the affine (not necessarily constant) current case, optimal trajectories can become straight lines, and in realistic routing or sail-assisted propulsion problems, strict convexity captures asymmetry and non-Euclidean kinetic constraints (Rossa et al., 15 Oct 2025).

7. Rigidity and Geodesic Structure in Projective and Hilbert Geometries

Strictly convex sets in projective and Hilbert geometries have uniquely geodesic structure: every boundary point admits a unique supporting hyperplane, and the Hilbert metric is strictly convex in the sense that every two points are connected by a unique geodesic (line segment). Any bijection preserving complete geodesics on such a set must be a projective transformation and, in the Hilbert geometry, an isometry (Chowdhury, 2022). For divisible sets with group actions, strict convexity, maximal anisotropy of boundary regularity, and simplicity of Lyapunov spectra are closely linked, implying strong rigidity and ergodic properties (Zimmer, 2013, Foulon et al., 2023).

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The strict convexity of control sets thus interweaves geometric, analytic, optimization, and dynamical considerations, enabling robust and unique optimal control, simplifying certification, supporting efficient computation, and underpinning the structural properties of the spaces and measures involved in advanced control and optimization applications.