Kernel-Based LMI Approaches to Solving the Hamilton-Jacobi-Bellman Equation and Nonlinear Optimal Control

We present a kernel-based linear matrix inequality (LMI) approach for the approximate solution of Hamilton--Jacobi--Bellman (HJB) equations arising in nonlinear optimal control. The method represents the gradient of the value function in a reproducing kernel Hilbert space (RKHS) and uses a Schur-complement reformulation to convert the quadratic HJB inequality into an LMI that is linear in the kernel coefficients, yielding a convex semidefinite program. The novel ingredient is an explicit Riccati--Hessian \emph{equality} constraint at the equilibrium, which removes the trivial solution and forces the Hessian of the approximation to match the algebraic Riccati equation solution of the linearised system. We give a suboptimality bound $J(x_0;\hat u) - V^*(x_0)\le \varepsilon\,T(x_0)$ in which $T(x_0)$ depends only on the problem data and the working domain (not on the approximation), and an RKHS approximation rate. Numerical experiments on a corrected 1D polynomial benchmark and on the Van der Pol oscillator measure $\varepsilon$, the RKHS approximation error, and the closed-loop cost $J(x_0;\hat u)$ versus the optimal value $V^*(x_0)$. On the 1D problem with $V^*$ in the polynomial-kernel RKHS the method recovers $V^*$ to within $3\times10^{-7}$ and achieves $0.000\%$ suboptimality. On Van der Pol it achieves the smallest HJB residual ($\varepsilon\approx 2.62$) of any method tested, beats LQR on every initial condition, and is within $0.42\%$ of the best per-IC cost (Albrekht order 6). When $V^*$ is not in the chosen RKHS, the method degrades gracefully: residuals stop improving with more centres but suboptimality remains bounded ($\le 13\%$ on the 1D test).