# Continuous Time Dynamic Programs

•  Continuous-time dynamic programs
• The HJB equation; a heuristic derivation; and proof of optimality.

Discrete time Dynamic Programming was given in previously (see Dynamic Programming ). We now consider the continuous time analogue.

Time is continuous $t\in\mathbb{R}_+$; $x_t\in \mathcal{X}$ is the state at time $t$; $a_t\in \mathcal{A}$ is the action at time $t$;

Def 1 [Plant Equation] Given function $f: \mathbb{R}_+\times\mathcal{X}\times \mathcal{A}_t \rightarrow \mathcal{X}$, the state evolves according to a differential equation This is called the Plant Equation.

Def 2 A policy $\pi$ chooses an action $\pi_t$ at each time $t$. The (instantaneous) reward for taking action $a$ in state $x$ at time $t$ is $r_t(a,x)$ and $r_T(x)$ is the reward for terminating in state $x$ at time $T$.

Def 3 [Continuous Dynamic Program] Given initial state $x_0$, a dynamic program is the optimization Further, let $C_\tau({\bf a})$ (Resp. $L_\tau(x_\tau)$) be the objective (Resp. optimal objective) for when the summation is started from $t=\tau$, rather than $t=0$.

When a minimization problem where we minimize loss given the costs incurred is replaced with a maximization problem where we maximize winnings given the rewards received. The functions $L$, $C$ and $c$ are replaced with notation $W$, $R$ and $r$.

Def 4 [Hamilton-Jacobi-Bellman Equation] For a continuous-time dynamic program , the equation is called the Hamilton-Jacobi-Bellman equation. It is the continuous time analogoue of the Bellman equation [[DP:Bellman]].

Ex 1 [A Heuristic derivation of the HJB equation] Argue that, for $\delta>0$ small, $x$ satisfying the recursion is a good approximation to the plant equation . (A heuristic argument will suffice)

Ex 2 [Continued] Argue (heuristically) that following is a good approximation for the objective of a continuous time dynamic program is Ex 3 [Continued]Show that the Bellman equation for the discrete time dynamic program with objective and plant equation is Ex 4 [Continued]Argue, by letting $\delta$ approach zero, that the above Bellman equation approaches the equation Ex 5 [Optimality of HJB] Suppose that a policy $\Pi$ has a value function $C_t(x,\Pi)$ that satisfies the HJB-equation for all $t$ and $x$ then, show that $\Pi$ is an optimal policy.

(Hint: consider $e^{-\alpha t}C_t(\tilde{x}_t,\Pi)$ where $\tilde{x}$ are the states another policy $\tilde{\Pi}$.)

Def 5. [LQ problem] We consider a dynamic program of the form Here $x_t \in\mathbb{R}^n$ and $a_t\in\mathbb{R}^m$. $A$ and $B$ arematrices. $Q$ and $R$ symmetric positive definite matrices. This an Linear-Quadratic problem (LQ problem).

Ex 6. [LQ Regulatization] Show that the HJB equation for an LQ problem is Ex 7. [Continued] We now “guess” that the solution to above HJB equation is of the form $L_t(x)=x^\top \Lambda(t) x$ for some symmetric matrix $\Lambda(t)$. Argue that the HJB equation is minimized by an action satisfying and thus for the minimum in the HJB equation to be zero we require Ex 8. [Continued] Show that for the HJB equation to be satisfied then it is sufficent that Def 6. [Riccarti Equation] The differential equation is called the Riccarti equation.

Ex 9. [Continued] Argue that a solution to the Riccarti Equation is optimal for the LQ problem.

Ans 1 Obvious from definition of derivative.

Ans 2 Obvious from definition of (Riemann) Integral and since $(1-\alpha \delta)^{t/\delta}\rightarrow e^{-\alpha t}$ as $\delta\rightarrow 0$.

Ans 3 Immediate from discrete time Bellman Equation.

Ans 4 Minus $L_t(x)$ from each side in  divide by $\delta$ and let $\delta\rightarrow 0$. Further note that Ans 5 Using shorthand $C=C_t(\tilde{x}_t,\Pi)$: The inequality holds since the term in the square brackets is the objective of the HJB equation, which is not maximized by $\tilde{\pi}_t$.

Ans 6. Immediate from Def 4.

Ans 7. $L_t(x) = x^\top \Lambda(x) x$ therefore Substituting into the Bellman equation gives Differentiating with respect to $a$ gives the optimality condition which implies Finally substituting into the Bellman equation, above, gives the required expression Ans 8. For the minimum in the HJB equation to be zero, we require Thus for the term in square brackets to be zero is sufficent.

Ans 9. This is just applying .