Theorems Convergence of Scaled Forward Euler Defect Multipliers to the Continuous Costate Attributions

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Statement

Let $T > 0$ and let $n,m \in \mathbb{N}$. Let $f \in C^2(\mathbb{R}^n \times \mathbb{R}^m;\mathbb{R}^n)$, $L \in C^2(\mathbb{R}^n \times \mathbb{R}^m;\mathbb{R})$, and $\Phi \in C^2(\mathbb{R}^n;\mathbb{R})$. Fix $x_0^* \in \mathbb{R}^n$ and consider the fixed-initial-state, free-terminal-state optimal control problem of minimizing

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\begin{align*} J[x,u]=\int_0^T L(x(t),u(t))\,d\mathcal{L}^1(t)+\Phi(x(T)) \end{align*}

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over trajectories $x \in C^1([0,T];\mathbb{R}^n)$ and controls $u \in C([0,T];\mathbb{R}^m)$ subject to

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\begin{align*} \dot{x}(t)=f(x(t),u(t)) \quad \text{for every } t\in[0,T] \end{align*}

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and

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\begin{align*} x(0)=x_0^*. \end{align*}

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Define the Pontryagin Hamiltonian $H: \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ by

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\begin{align*} H(x,p,u)=L(x,u)+p\cdot f(x,u). \end{align*}

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Suppose that $(x^*,u^*,p^*) \in C^1([0,T];\mathbb{R}^n) \times C([0,T];\mathbb{R}^m) \times C^1([0,T];\mathbb{R}^n)$ is a normal extremal satisfying, for every $t \in [0,T]$,

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\begin{align*} \dot{x}^*(t)=f(x^*(t),u^*(t)). \end{align*}

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\begin{align*} -\dot{p}^*(t)=\nabla_x H(x^*(t),p^*(t),u^*(t)). \end{align*}

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\begin{align*} 0=\nabla_u H(x^*(t),p^*(t),u^*(t)). \end{align*}

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\begin{align*} p^*(T)=\nabla\Phi(x^*(T)). \end{align*}

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Assume that there exists $\alpha>0$ such that the symmetric Hessian matrix $\nabla_{uu}^2H(x^*(t),p^*(t),u^*(t))\in\mathbb{R}^{m\times m}$ satisfies

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\begin{align*} \xi\cdot \nabla_{uu}^2H(x^*(t),p^*(t),u^*(t))\xi\ge \alpha |\xi|^2 \quad \text{for every } t\in[0,T] \text{ and every } \xi\in\mathbb{R}^m. \end{align*}

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Assume also that the second variation of $J$ at $(x^*,u^*)$ is coercive on the continuous critical cone for the fixed-initial-state problem.

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Let $(\mathcal{T}_h)$ be a family of meshes

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\begin{align*} \mathcal{T}_h=\{0=t_0<t_1<\cdots<t_N=T\}, \end{align*}

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where $N=N(h)$ may depend on $h$. Define $h_k=t_{k+1}-t_k$ for $0 \le k \le N-1$ and $h=\max_{0\le k\le N-1} h_k$. Assume that the meshes are quasi-uniform: there exists $\rho>0$, independent of $h$, such that $h_k \ge \rho h$ for all $0 \le k \le N-1$.

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For each mesh, consider the normalized forward Euler transcription over variables $x_k\in\mathbb{R}^n$ for $0\le k\le N$ and $u_k\in\mathbb{R}^m$ for $0\le k\le N-1$:

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\begin{align*} \min_{(x_k,u_k)}\left\{\sum_{k=0}^{N-1}h_kL(x_k,u_k)+\Phi(x_N)\right\} \end{align*}

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subject to

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\begin{align*} x_0=x_0^*. \end{align*}

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\begin{align*} \frac{x_{k+1}-x_k}{h_k}-f(x_k,u_k)=0 \quad \text{for } 0\le k\le N-1. \end{align*}

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Assume that for all sufficiently small $h$ there exists a locally optimal feasible point $(x_{k,h},u_{k,h})$ satisfying

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\begin{align*} \max_{0\le k\le N}|x_{k,h}-x^*(t_k)|+\max_{0\le k\le N-1}|u_{k,h}-u^*(t_k)| \to 0 \end{align*}

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as $h \to 0$. Assume that the equality-constraint LICQ holds at $(x_{k,h},u_{k,h})$ for the variables $(x_1,\dots,x_N,u_0,\dots,u_{N-1})$ with $x_0=x_0^*$ fixed, so that the defect multiplier vector is unique.

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Let $\lambda_{k,h}\in\mathbb{R}^n$ denote the multiplier attached to the normalized defect constraint

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\begin{align*} \frac{x_{k+1}-x_k}{h_k}-f(x_k,u_k)=0 \end{align*}

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in the discrete Lagrangian

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\begin{align*} \mathcal{L}_h=\sum_{k=0}^{N-1}h_kL(x_k,u_k)+\Phi(x_N)+\sum_{k=0}^{N-1}\lambda_{k,h}\cdot\left(\frac{x_{k+1}-x_k}{h_k}-f(x_k,u_k)\right). \end{align*}

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Define the recovered discrete costate $\mu_{k,h}\in\mathbb{R}^n$ by

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\begin{align*} \mu_{k,h}=-h_k^{-1}\lambda_{k,h} \quad \text{for } 0\le k\le N-1. \end{align*}

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Assume the following mesh-independent maximum-norm stability estimate for the discrete adjoint terminal-value equations. For every sufficiently small $h$, define $A_{k,h}\in\mathbb{R}^{n\times n}$ for $1\le k\le N-1$ by

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\begin{align*} A_{k,h}=\nabla_x f(x_{k,h},u_{k,h})^\top. \end{align*}

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For every family of vectors $r_{k,h}\in\mathbb{R}^n$ for $1\le k\le N-1$, every terminal vector $b_h\in\mathbb{R}^n$, and every sequence $z_{k,h}\in\mathbb{R}^n$ for $0\le k\le N-1$ satisfying

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\begin{align*} z_{k-1,h}-z_{k,h}-h_k A_{k,h}z_{k,h}=r_{k,h} \quad \text{for } 1\le k\le N-1, \end{align*}

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and

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\begin{align*} z_{N-1,h}=b_h, \end{align*}

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there exists a constant $C>0$, independent of $h$, $r_{k,h}$, $b_h$, and $z_{k,h}$, such that

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\begin{align*} \max_{0\le k\le N-1}|z_{k,h}|\le C\left(|b_h|+\max_{1\le k\le N-1}|r_{k,h}|\right). \end{align*}

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Then

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\begin{align*} \max_{0\le k\le N-1}\left|-h_k^{-1}\lambda_{k,h}-p^*(t_k)\right|\to0 \end{align*}

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as $h\to0$.

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