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Neural Time-Reversed Generalized Riccati Equation
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Proposes a forward-in-time neural method for optimal control that estimates costates with a network whose parameters evolve via a time-reversed generalized Riccati equation, conjectured to stabilize Hamilton equations.
Problem
Optimal control via Hamilton equations requires forward-backward shooting over the whole time horizon, which is computationally infeasible for high-dimensional states or long horizons, and Hamilton equations are generally unstable.
Approach
Estimate the costate with a feed-forward network whose parameters are updated locally at each time step by a time-reversed generalized Riccati equation (derived from the scalar LQ case), enabling forward-only integration of state, network weights and costate; the policy is conjectured to stabilize the Hamiltonian dynamics.
Key results
- In the scalar LQ example, the time-reversed Riccati equation with zero initial condition converges to the infinite-horizon optimal solution lambda_1 R/B^2 as T and s grow.
- The conjecture that the time-reversed generalized Riccati policy stabilizes Hamilton equations is supported by experiments on several optimal control case studies (details not in the provided text).
Surprising findings
- Hamilton equations for the scalar LQ problem have positive exponential modes, so the forward costate dynamics are unstable.
Limitations
- Stability is only conjectured, not proven, for the general case.
- Theory is derived from the scalar LQ example and the method assumes costate is a function of state.
- Experimental details are deferred to an external input file and are not included in the provided text.
Experimental setup
Builds on: Pontryagin Maximum Principle Hamilton-Jacobi-Bellman equation Continuous-Time Recurrent Neural Networks Linear Quadratic regulator
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