Maximum Energy Shapeability

• Maximum Energy Shapeability is the principle of configuring geometric shapes to maximize an energy functional under physical constraints.
• It bridges discrete and continuous optimization by integrating convex geometry, PDE analysis, and energy shaping across diverse engineering systems.
• This concept yields practical strategies for designing optimal configurations in materials, control systems, and electromagnetic as well as hydrodynamic applications.

Maximum Energy Shapeability is a unified concept spanning mathematical, physical, and engineering domains, concerned with the interplay between geometric configuration and the maximum realizable energetic functional under a set of constraints. Its intellectual development connects discrete and continuous optimization, the geometry of polygons and curves, elastic materials with distributed memory, passivity-based control in nonlinear systems, electromagnetic field engineering, and hydrodynamic shape optimization. The central theme is the determination and exploitation of geometric shape—or configuration—that maximizes a specific energy or functional (often subject to constraints such as non-extensibility, boundary conditions, or actuation limits).

1. Foundational Definitions and Mathematical Formalism

The archetype of maximum energy shapeability is the problem of point distributions on non-extensible closed curves in $\mathbb{R}^3$: for a cyclic polygonal chain $G_n$ of length $n$, with $n$ points $\{A_1, \dots, A_n\}$ evenly distributed by arc length, and a strictly increasing continuous function $f: (0,\infty)\to\mathbb{R}$, the total energy is

$E^f_n(G_n) = \sum_{1\leq p<q\leq n} f(|A_pA_q|),$

where $|A_pA_q|$ is the Euclidean distance between points $A_p$ and $A_q$ (Cheng et al., 2023). The maximum energy shapeability problem asks for the shapes $G_n$ that maximize $E^f_n(G_n)$ under the constraint that consecutive points are unit distance apart (i.e., edges of length $1$).

A paradigmatic example is the power law $f_\alpha(t) = t^\alpha$, $0<\alpha\leq 2$, which interpolates between mild and strong long-range reward for distant pairs. The definition generalizes to systems such as elastic spring lattices, underactuated control systems, electromagnetic fields, and shape-optimized physical domains, each with its own extensions to the notion of shapeability and maximizeable functionals.

2. Principal Results in Discrete Geometric Energy Maximization

For the polygonal curve model, Cheng and Wang established fundamental theorems on maximizers (Cheng et al., 2023):

• Existence: There always exists a maximizer $G_n$ for any continuous, increasing $f$.
• Planarity and Convexity: All maximizers are planar convex $n$-gons with edges of length $1$; nonplanar configurations can always be reflected and unfolded to increase energy.
• Uniqueness and Phase Transition: For $n=4$ (quadrilaterals), the maximizer switches sharply (“phase transition”) from a square ($\alpha<2$) to two degenerate “double arc” configurations for $\alpha>2$.
• Regular Polygon Maximization: For $n\geq 5$ and $0<\alpha\leq 2$, the unique maximizer is the regular $n$-gon of unit edges.

The characterization is supported by majorization arguments, Jensen’s inequality (for concave $f$), and kinetic energy (moment of inertia) constraints for the quadratic case $\alpha=2$. The energy decomposes into $k$-step chord contributions, and convexity/planarity is enforced by reflection and “fold-along-an-edge” procedures.

This framework formalizes the maximum energy shapeability principle: for concave energy rewards ($\alpha\leq 2$) maximal energy is achieved by uniform (regular) planar distribution, while for convex rewards ($\alpha>2$) the system undergoes symmetry breaking, often coalescing into maximal-diameter (degenerate) configurations.

3. Maximum Energy Shapeability in Control Systems and Passivity-Based Design

The notion of maximum energy shapeability has been adopted within control theory, particularly in the context of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) for affine nonlinear systems (Jiao et al., 4 Jan 2026). Here, the “shape” refers to the energy landscape engineered via the assignment of a desired Hamiltonian $H_d(x)$.

Given the matching PDE

$G^\perp(x) F_d(x) \nabla H_d(x) = G^\perp(x) f(x),$

where $x\in\mathbb{R}^n$, $u\in\mathbb{R}^m$, and $G$ has rank $m<n$, maximum energy shapeability occurs when the homogeneous equation

$G^\perp(x) F_d(x) \nabla H_d(x) = 0$

admits $m$ independent (smooth) solutions—i.e., the maximal number allowed by the input dimension.

Sufficient conditions for maximum shapeability include:

• The Poincaré condition: each $F_d^{-1}g_i$ is a gradient field.
• For a single underactuated degree ($m=n-1$), maximum shapeability always holds.
• For constant coefficient systems, the integrability condition $L_{w_i}s_j - L_{w_j}s_i\equiv 0$ ensures solvability and maximal shapeability.

The constructive procedure under maximal shapeability involves:

1. Solving for $m$ characteristic coordinates (homogeneous solutions).
2. Changing variables to decouple the PDE.
3. Solving reduced, simpler PDEs in the remaining directions.
4. Enforcing integrability and definiteness by solving for a symmetric matrix of partials.
5. Reconstructing the solution and implementing the final control law.

This structure underpins many constructive IDA-PBC methods and allows closed-form or systematically simplified controller synthesis for broad classes of nonlinear systems.

4. Material and Physical Systems: Elastic Sheets and Shape-Memory Networks

In physical systems, maximum energy shapeability is realized through materials that can be programmed or imprinted to achieve multiple stable shapes. Oppenheimer and Witten analyze two-dimensional triangular spring lattices where randomness or periodic frustration in rest-lengths induces a rugged energy landscape with many metastable minima (Oppenheimer et al., 2015).

Key findings include:

• Shape memory arises from bistability in node configurations, with 30–40% bistable nodes in random $8\times8$ lattices.
• Hysteretic transitions between shapes are cooperative, typically involving 12 out of 49 nodes.
• The energetic barriers governing shape retention scale as $\frac{1}{2}k(\Delta \ell)^2$; design strategies include hierarchical thickness (multi-scale rest-length patterns) or lowering bistability thresholds to promote shape imprinting at smaller curvatures.
• Sheet size controls memory retention nonmonotonically, with an optimal regime at moderate $N$.

In these materials, maximum energy shapeability is directly connected to the richness of the energy landscape: tuning the network produces $2^{O(N)}$ local minima, enabling diverse and robust memory behavior.

5. Shape Optimization in Electromagnetic and Hydrodynamic Energy Functionals

In electromagnetic design, maximum energy shapeability addresses the optimal concentration of electric energy in a user-specified subregion, subject to power constraints (Teuwen et al., 2018). The maximization process:

• Is formulated as an eigenproblem for a compact Hermitian integral operator whose kernel encodes the geometry of the target region.
• Involves expanding the field in angular spectra and imposing power normalization.
• The optimal field distribution achieves maximal concentration by matching the eigenfield of the largest eigenvalue.
• In lens systems, this theoretical optimum can be realized by shaping the amplitude, phase, and polarization of the entrance pupil using spatial light modulators.

Hydrodynamic systems exhibit analogous principles. The optimization of the shape of a planar wave energy converter (WEC) for maximal absorbed power over a spectrum of incident waves is approached by:

• Parametrizing candidate shapes via Fourier modes and elongation factors.
• Employing a genetic algorithm to search the high-dimensional space of shapes, using hydrodynamic coefficients (added mass, damping, excitation) computed with a BEM solver.
• Tuning power take-off parameters for each candidate shape.
• Significant enhancement in absorbed energy is achieved—e.g., 250–300% over circular geometry for unidirectional waves, with elongated “racetrack” and multi-wing “butterfly” forms emerging as optimal under different incident field conditions (Esmaeilzadeh et al., 2018).

6. Theoretical and Practical Implications

Maximum energy shapeability integrates principles of convex geometry, PDE analysis, group symmetries, and functional analysis in infinite- and finite-dimensional spaces. Its significance includes:

• Symmetry breaking and phase transitions: For sufficiently convex reward functions, symmetry maximizers give way to degenerate, concentrated configurations—analogous to phase transitions.
• Design of materials and control: The systematic exploitation of shapeability enables programmable matter and tractable nonlinear controller design via decoupling of the matching PDE.
• Interconnection with optimization and computational methods: High-dimensional search procedures (genetic algorithms, finite-difference solvers, adjoint-state gradients) leverage the geometry–energy relationship for practical engineering devices.
• Scalable and hierarchical architectures: Materials or control systems with multi-scale or compositional structure can be designed for desired shape/memory properties by embedding hierarchical frustration or symmetry.

A central point is that shapeability is governed by the allowed degrees of freedom subject to system-imposed constraints and the superlinear (convex) or sublinear (concave) nature of the energetic reward for separation or geometric extension. The critical domain is the regime where the system’s “shape space” is just rich enough to allow the maximal number of independent “energy shaping” directions.

7. Connections Across Domains and Open Problems

Maximum energy shapeability bridges mathematical analysis (e.g., polygonal energy maximization (Cheng et al., 2023)), materials science (memory-rich lattices (Oppenheimer et al., 2015)), nonlinear control (IDA-PBC (Jiao et al., 4 Jan 2026)), electromagnetic field design (Teuwen et al., 2018), optimization for light–matter interactions (Matuszak et al., 2022), and hydrodynamics (Esmaeilzadeh et al., 2018). Open challenges include:

• Rigorous classification of critical exponents (e.g., the value $\alpha^*_n$ for phase transitions in geometric maximization).
• Generalization to higher dimensions or to systems with additional topological constraints.
• Synthesis of materials with programmable, robust multi-stability at macroscopic scales.
• Extension of the control-theoretic conditions for maximum shapeability to classes of underactuated distributed systems.
• Efficient computational strategies for shape optimization in domains with complex dynamical interactions.

Maximum energy shapeability thus defines a central organizing principle in the structure–function landscape, illuminating how physical, geometric, and informational constraints conspire to determine—and often optimize—the realized energetic output of complex systems.