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Continuous adjoint equation for the linear-quadratic ODE
Definition
For the data of Finite-horizon bounded-control linear-quadratic ODE problem and an admissible control with state , a continuous LQR adjoint is a curve whose coordinates are absolutely continuous in the scalar sense of Absolute continuity on a compact interval, satisfying, almost everywhere,
The equation defines the adjoint notion without asserting existence. Existence and uniqueness for each admissible control are established under Countable Choice by The backward LQR adjoint equation has a unique solution ↗. This sign convention is the one for the minimization Hamiltonian .
Depends on
- Finite-horizon bounded-control linear-quadratic ODE problem
- The bounded measurable linear ODE has a unique affine state map
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- Absolute continuity on a compact interval
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
Used by
- An active terminal constraint invalidates the pointwise control-only variational inequality Counterexample
- A bounded-actuator LQR control and its exact objective Example
- The backward LQR adjoint equation has a unique solution Lemma
- The LQR adjoint identity gives the reduced gradient Lemma
- LQR minimizers satisfy a feasible-set variational inequality Theorem
Dependency tree · two levels
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Sources
- J. Monnier, Optimal Control of ODEs: The Linear-Quadratic (LQ) Case, course notes, Chapter 2 §2.5.2 (standard reference, not scraped)
- W. W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, Numerische Mathematik 87 (2000) (standard reference, not scraped)