Dual Variational Formulation in Quadratic Systems

• Dual variational formulation is a framework that reformulates traditionally non-convex quadratic system analysis into convex optimization problems using dual variables and penalization techniques.
• The approach employs methods such as Legendre transforms, saddle-point interchanges, and Fenchel duality to robustly address both ODEs like the Quadratic–Quadratic Regulator and Hamiltonian PDEs.
• This framework overcomes limitations of classic Riccati methods by enabling existence, uniqueness, and efficient numerical schemes for large-scale, time-inhomogeneous, or measure-valued systems.

A dual variational formulation for quadratic systems refers to a family of mathematical strategies that restate the analysis, control, or solution of systems governed by quadratic nonlinearities—whether ODEs or PDEs—as convex optimization problems posed in dual variables. These formalisms enable reformulation of traditionally non-convex, often Riccati-based, and potentially high-dimensional quadratic control or evolution problems into convex, sometimes quadratic-growth, problems solvable by standard convex optimization approaches. This paradigm spans optimal control, Hamiltonian PDEs (such as Nash systems or Hamilton–Jacobi equations), and systems with quadratic dynamics or output structures, bridging classical theory and large-scale numerics through dualization, penalization, and exploitation of hidden convexity.

1. General Principles of Dual Variational Formulation

The dual variational approach constructs an optimization problem in variables conjugate (dual) to the original (primal) system variables. For quadratic dynamics or cost structures, these functionals can frequently be made (strongly) convex, with the stationarity conditions of the dual functional directly encoding necessary optimality or dynamical evolution conditions for the primal system.

In the context of deterministic finite-dimensional ODEs, such as the Quadratic–Quadratic Regulator (QQR)—an extension of Linear–Quadratic Regulator (LQR) to include state-dependent quadratic nonlinearities—the starting point is the Pontryagin Maximum Principle (PMP), with its adjoint-based optimality conditions. For quadratic Hamiltonian PDEs (including the Nash system or the quadratic Hamilton–Jacobi equation), dual formulations encode weak or measure-valued solutions and systematically address potential non-smoothness, shocks, or absence of a classical viscosity framework (Acharya et al., 21 Feb 2025, Vorotnikov et al., 14 Dec 2025).

Key steps across these domains involve:

• Introduction of dual variable paths (e.g., Lagrange multipliers, adjoint fields) to enforce constraints or encode dynamics.
• Penalization schemes to ensure strict convexity and coercivity, enabling existence and uniqueness results in suitable function spaces.
• Formulation of dual functionals via Legendre transform, saddle-point (min–max) interchanges, or Fenchel duality.

2. Quadratic-Quadratic Regulator and ODE Systems

The dual formulation developed for the QQR problem involves augmenting the standard PMP Hamiltonian with quadratic penalty terms:

$$Q(x,u,p)=\tfrac12 (a_x|x-\bar x|^2 + a_u|u-\bar u|^2 + a_p|p-\bar p|^2),$$

where $(\bar x, \bar u, \bar p)$ serves as a guiding trajectory. The resulting pre-dual functional,

$\widehat S_Q[x,u,p,\gamma,\mu,\lambda],$

is affine in the dual variables $(\gamma, \mu, \lambda)$ and strictly concave in the primal variables $(x, u, p)$. Stationarity in the primal variables leads to linear equations encoding the PMP system, while criticality in the dual variables enforces boundary and transversality conditions.

Passing to the dual (Legendre) form yields the convex dual functional,

$$\tilde S_Q[\gamma, \mu, \lambda]=\sup_{x,u,p} \widehat S_Q[x,u,p,\gamma,\mu,\lambda],$$

whose minimizers can be shown to exist by coercivity and lower semicontinuity in the Hilbert space

$$(\gamma, \lambda) \in H^1(0,T;\mathbb{R}^n),\quad \mu \in L^2(0,T;\mathbb{R}^m).$$

In the linear–quadratic special case ($F\equiv 0$), the structure further simplifies and uniqueness follows. The dual functional is a convex boundary-value problem in dual variables, directly encoding the PMP equations (Acharya et al., 21 Feb 2025).

This approach contrasts with the classic Riccati ODE-based method, offering a pathway to convexify the solution of large-scale or time-inhomogeneous quadratic control problems through standard convex optimization, sidestepping inherent nonlinearities and opening avenues for robust numerical frameworks.

3. Variational Duality in Quadratic Hamiltonian PDEs

For first-order quadratic Hamiltonian PDEs, such as the Nash system,

$$\partial_t \psi + \mathcal{O}(\nabla \psi \otimes \nabla \psi) = 0,$$

dual variational functionals are derived using weak or measure-valued notions of solutions. The primal kinetic energy

$$E_{\mathrm{rel}}[v;\bar v] = \frac12 \int_0^T \|v(t) - \bar v(t)\|_2^2\,dt$$

is minimized over weak solutions $v$, relative to a chosen base state $\bar v$. A saddle-point (pre-dual) formulation couples $v$ to dual fields $E, B$ under linear constraints. Interchanging inf and sup yields a convex dual maximization problem, constrained by positive semidefiniteness and auxiliary linear relations,

$$\text{maximize} \ S[E,B;\bar v] \quad \text{subject to}\ \partial_t B + \mathcal{O}^* E = 0,\ I + 2B \geq 0,$$

where $v = (I+2B)^{-1}(E - \bar v)$ (Vorotnikov et al., 14 Dec 2025).

Smooth dual maximizers yield stationarity equations that recover the original PDE in “DtP” (dual-to-primal) mapping form. In essence, the dual problem realizes the primal quadratic flow as the Euler–Lagrange equation for a convex functional, emphasizing the “hidden” convexity of a classically nonlinear problem.

4. Role and Construction of Base States

A central feature in dual variational formulations for quadratic systems is the selection of base states (guiding trajectories or fields) $\bar v$. Their twofold role is:

• Ensuring strict convexity and coercivity of the dual functional within a neighborhood of the minimizer or maximizer,
• Facilitating consistency (zero duality gap), that is, guaranteeing that the primal solution can be exactly reconstructed from the dual optimal variables.

A sufficient condition for global-in-time consistency is the existence of a symmetric-matrix field $G$ and vector field $u$ such that

$$G \geq 0,\quad \partial_t(G-I)+2\mathcal{O}^* (Gu)=0,\quad v = G(v-u).$$

Setting $\bar v = G(v-u)$ ensures that the dual maximizer reconstructs the entire primal solution. For the Hamilton–Jacobi equation (scalar case), sequences of “small” consistent base states converging to zero in mean can be constructed, enabling dual recovery of solutions even as the time interval grows. This feature addresses the duality gap issue arising with naive zero base-state selection over large intervals (Vorotnikov et al., 14 Dec 2025).

5. Existence, Uniqueness, and Numerical Schemes

For a broad class of quadratic PDEs with trace-class operators $\mathcal{O}$ (including Burgers-type systems), the dual functional is convex and upper semicontinuous over suitable spaces of dual variables,

$$E \in L^2((0,T) \times \Omega;\mathbb{R}^n),\quad B \in L^\infty((0,T)\times\Omega;\mathbb{R}^{n \times n}_s).$$

Existence of dual solutions is established via coercivity, boundedness (trace condition for $I+2B \geq 0$), and weak-* compactness arguments. For non-scalar systems, unique dual maximizers may not exist classically, but existence is always guaranteed in the measure or $L^2$ sense (Vorotnikov et al., 14 Dec 2025).

An adaptive convex gradient-flow scheme iteratively progresses by updating the base state and solving a convex optimization in the dual, ensuring convergence (in practice, in a few stages) to a primal weak solution. This approach is robust to poor initial guesses and mitigates ill-conditioning, providing practical means for numerical solution of otherwise intractable quadratic evolution problems.

6. Related Duality Formulations: Quadratic Output and Weak PDE Solutions

In bilinear systems with quadratic outputs, variational dual formulations are instrumental for model reduction and Gramian computation. Here, primal–dual energy functionals induce Lyapunov or generalized Lyapunov equations for system Gramians, whose solution structures and existence criteria mirror those found in quadratic-state dual problems (Faßbender et al., 4 Jul 2025).

Similarly, for nonlinear quadratic porous medium equations (QPME), dual variational functionals arise either via a saddle formulation (min–max over primal and dual test functions) or a fully dual formulation:

$$J_{\text{dual}}(u_0) := \sup_{\phi \in B} \int_Q \left[-\frac{(\partial_t\phi)^2}{1-\Delta\phi} + 2u_0\partial_t\phi\right] dx\,dt,$$

subject to convexity-type constraints. Such duality connects entropy minimization in weak solution spaces with convex maximization principles, providing existence, uniqueness, and convergence guarantees (Lu et al., 2022).

7. Significance, Generalizations, and Comparative Analysis

The dual variational framework for quadratic systems consolidates optimal control, nonlinear PDEs, and system-theoretic model reduction under a unified convex-analytic perspective. It systematically replaces non-convex or nonlinear Riccati and Hamilton–Jacobi approaches with convex optimization subject to quadratic growth and linear constraints in dual variables. Applications range from nonlinear regulators and Nash systems to model reduction of weakly nonlinear dynamical systems.

Compared to classical methods, dual variational formulations:

• Enable use of large-scale convex solvers (finite element, spectral, gradient flow) without hidden nonlinearities,
• Handle time-dependent forcing, degenerate cost matrices, and non-conservative or measure-valued regimes,
• Offer robust theoretical guarantees (existence/coercivity), and
• Generalize to accommodate system, matrix, or weak measure-valued solutions.

A plausible implication is the foundational shift toward convex optimization as a standard computational and analysis tool for quadratic systems beyond strictly linear contexts (Acharya et al., 21 Feb 2025, Faßbender et al., 4 Jul 2025, Vorotnikov et al., 14 Dec 2025, Lu et al., 2022).