Pontryagin minimum principle

For every pair of natural numbers $n,m$ (including $0$), let the state and control spaces be the real Euclidean spaces $ℝ^n$ and $ℝ^m$. Fix real numbers $t₀<t₁$; maps $F:ℝ^n→ℝ^m→ℝ^n$ and $ℓ:ℝ^n→ℝ^m→ℝ$; proposed state derivatives $Fₓ(x,u):ℝ^n→Lℝ^n$ and $ℓₓ(x,u):ℝ^n→Lℝ$; an arbitrary set $Ω⊆ℝ^m$; an initial state $x_{\mathrm{Init}}∈ℝ^n$; and total paths $u₀:ℝ→ℝ^m$ and $x₀:ℝ→ℝ^n$. Assume, for every state $x$ and control $u$, that $Fₓ(x,u)$ is the Fréchet derivative at $x$ of $y↦F(y,u)$ and $ℓₓ(x,u)$ is the Fréchet derivative at $x$ of $y↦ℓ(y,u)$; assume that the four jointly curried maps $(x,u)↦F(x,u)$, $(x,u)↦ℓ(x,u)$, $(x,u)↦Fₓ(x,u)$, and $(x,u)↦ℓₓ(x,u)$ are continuous; and assume that $t↦Fₓ(x₀(t),u₀(t))$ and $t↦ℓₓ(x₀(t),u₀(t))$ are interval-integrable with respect to Lebesgue volume from $t₀$ to $t₁$. Assume also that there is a real constant $C$, with no separately stated condition $C≥0$, such that $‖u₀(t)‖≤C$ for Lebesgue-almost every $t$ in the closed interval $[t₀,t₁]$, and that there is a real $M≥0$ such that, for all states $x,y$ and controls $u,v$, $‖F(x,u)-F(y,v)‖≤M(‖x-y‖+‖u-v‖)$. Finally assume that $(u₀,x₀)$ is globally optimal in the following fully specified sense: $x₀(t₀)=x_{\mathrm{Init}}$; $x₀$ is absolutely continuous on $[t₀,t₁]$; $u₀$ is strongly measurable modulo a null set for Lebesgue measure restricted to $[t₀,t₁]$; $u₀(t)∈Ω$ for almost every such $t$; $x₀$ is differentiable with $x₀'(t)=F(x₀(t),u₀(t))$ for almost every $t∈(t₀,t₁)$; $t↦ℓ(x₀(t),u₀(t))$ is interval-integrable; and, for every pair of total paths $u:ℝ→ℝ^m$ and $x:ℝ→ℝ^n$ satisfying those same admissibility conditions—namely $x(t₀)=x_{\mathrm{Init}}$, absolute continuity of $x$ on $[t₀,t₁]$, almost-everywhere strong measurability of $u$ there, $u(t)∈Ω$ almost everywhere there, $x'(t)=F(x(t),u(t))$ almost everywhere on $(t₀,t₁)$, and interval-integrability of $t↦ℓ(x(t),u(t))$—one has $∫_{t₀}^{t₁}ℓ(x₀(t),u₀(t))\,dt≤∫_{t₀}^{t₁}ℓ(x(t),u(t))\,dt$. Then there exists a total path $λ:ℝ→ℝ^n$ such that $λ(t₁)=0$, $λ$ is absolutely continuous on $[t₀,t₁]$, and, for Lebesgue-almost every $t∈(t₀,t₁)$, there exists a vector $dλ∈ℝ^n$ for which $λ$ is differentiable at $t$ with derivative $dλ$ and, for every $h∈ℝ^n$, $⟨-dλ,h⟩=⟨λ(t),Fₓ(x₀(t),u₀(t))(h)⟩+ℓₓ(x₀(t),u₀(t))(h)$. Moreover, for Lebesgue-almost every $t∈[t₀,t₁]$ and every $v∈Ω$, $H(x₀(t),u₀(t),λ(t))≤H(x₀(t),v,λ(t))$, where $H(x,u,λ)=⟨λ,F(x,u)⟩+ℓ(x,u)$. Each separately stated almost-everywhere condition may have its own exceptional null set. In particular, neither the state equation nor the adjoint identity is asserted at an endpoint, and the control bound, membership in $Ω$, and Hamiltonian comparison are phrased over the closed interval only almost everywhere, so they need not hold at either singleton endpoint; the exact endpoint assertions are $x₀(t₀)=x_{\mathrm{Init}}$ and $λ(t₁)=0$. No uniqueness or nonzero condition is imposed on $λ$, and the theorem imposes no conditions on the paths outside the interval beyond their being total functions. Although $Ω$ is formally arbitrary, if $Ω=∅$ then the optimal-pair hypothesis is unsatisfiable on the positive-length interval, so that instance is vacuous. The zero-dimensional state and control cases remain included.