Motivation

Dynamic programming is the backbone of stochastic optimal control, operations research, and reinforcement learning. Its cornerstone — that the backward recursion of Bellman computes the optimal cost of a finite-horizon stochastic control problem — is stated as Proposition 1.3.1 of Bertsekas, Dynamic Programming and Optimal Control, Vol. I (3rd ed., Athena Scientific, 2005), the standard graduate text on the subject. Every convergence result for value iteration, every performance bound for approximate DP, and every correctness proof for a planning algorithm ultimately leans on this proposition. A machine-checked version of it — over a clean, reusable model of the basic problem — is the natural foundation stone for formalized control theory and RL theory alike.

Setting

The basic problem (§1.2 of the book): a discrete-time system

$$x_{k+1} = f_k(x_k, u_k, w_k), \qquad k = 0, 1, \dots, N-1,$$

with state $x_k \in S$, control $u_k$ constrained to a finite nonempty set $U_k(x_k) \subseteq C$, and disturbance $w_k$ drawn from a finite space $W$ with conditional probabilities $p_k(w \mid x_k, u_k)$. A policy is a sequence $\pi = \{\mu_0, \mu_1, \dots\}$ of feedback maps $\mu_k : S \to C$; it is admissible if $\mu_k(x) \in U_k(x)$ everywhere. Its expected cost from $x_0$ is

$$J_\pi(x_0) = \mathbb{E}\Big[ g_N(x_N) + \sum_{k=0}^{N-1} g_k(x_k, \mu_k(x_k), w_k) \Big].$$

In the Lean development these are BertsekasDPModel, BertsekasDPPolicyCost (backward recursion on remaining stages), and the DP recursion BertsekasDPValue:

$$J_N = g_N, \qquad J_k(x) = \min_{u \in U_k(x)} \mathbb{E}_w\big[ g_k(x,u,w) + J_{k+1}(f_k(x,u,w)) \big].$$

Section 1.6 of the book develops the minimax variant, where the disturbance is chosen antagonistically from a finite membership set $W_k(x,u)$; the mission mirrors it with BertsekasMinimaxDPModel, BertsekasMinimaxPolicyCost, BertsekasMinimaxValue.

Target

$$J_0(x_0) \;=\; \min_{\pi \text{ admissible}} J_\pi(x_0), \qquad \text{with the minimum attained,}$$

formalized as BertsekasDP.dp_algorithm_optimality: the DP value at the horizon is an IsLeast of the set of admissible policy costs. Milestones: the min–max interchange Lemma 1.6.1 (minimax_selection_interchange) and the minimax DP validity (minimax_dp_algorithm).

Significance

The proposition itself is the license to compute optimal policies stage by stage; downstream, Missions VI and VII of this series (lookahead bounds, infinite-horizon theory) consume exactly this model and recursion. Formalizing it produces the reusable model of the basic problem — the shared vocabulary for the whole series. The result is classical and proved in the book; the contribution here is a machine-checked proof over a model faithful to the book's, with the measurable-selection subtleties deliberately avoided by finiteness (see scope).

Difficulty

The proof is a backward induction, but the standard informal argument ("interchange expectation and minimization") must be carried out honestly: the induction hypothesis is about all states simultaneously, the minimizing control must be selected as a function of the state (choice over a finite set), and the policy-cost recursion must be related to the value recursion stage by stage. The minimax milestone needs the interchange lemma with its $> -\infty$ proviso — the classic trap is losing that hypothesis and asserting a false unconditioned interchange.

Formalization scope

Finite disturbance space (Fintype W), finite nonempty control-constraint sets (Finset, inf'), arbitrary (possibly infinite) state space; expectations are finite weighted sums, probabilities are required to be distributions only at admissible controls. Stage data are total functions on $\mathbb{N}$; only stages $0,\dots,N-1$ matter. Policies are deterministic Markov feedback maps — for this class the book's result is exactly recovered. The trivializing risks (empty constraint sets, junk beyond horizon) are ruled out by the nonemptiness field and by evaluating at exactly $N$ remaining stages. Lemma 1.6.1 is stated in the extended reals over arbitrary types with the book's finiteness-of-infimum proviso.

Selected references

• D. P. Bertsekas, Dynamic Programming and Optimal Control, Vol. I, 3rd ed., Athena Scientific, 2005. ISBN 1-886529-26-4. (Prop. 1.3.1, §1.2–1.3, §1.6.) http://www.athenasc.com/dpbook.html
• R. Bellman, Dynamic Programming, Princeton University Press, 1957.