An adaptive mesh refinement method for indirectly solving optimal control problems

Abstract

The indirect solution of optimal control problems (OCPs) with inequality constraints and parameters is obtained by solving the two-point boundary value problem (BVP) involving index-1 differential-algebraic equations (DAEs) associated with its first-order optimality conditions. This paper introduces an adaptive mesh refinement method based on a collocation method for solving the index-1 BVP-DAEs. The paper first derives a method to estimate the relative error between the numerical solution and the exact solution. The relative error estimate is then used to guide the mesh refinement process. The mesh size is increased when the estimated error within a mesh interval is beyond the numerical tolerance by either increasing the order of the approximating polynomial or dividing the interval into multiple subintervals. In the mesh interval where the error tolerance has been met, the mesh size is reduced by either decreasing the degree of the approximating polynomial or merging adjacent mesh intervals. An efficient parallel implementation of the method is implemented using Python and CUDA. The paper presents three examples which show that the approach is more computationally efficient and robust when compared with fixed-order methods.

Appendix: A

1.1 A.1 Derivation of the necessary conditions

To present the necessary conditions for a minimum, define the scalar Hamiltonian function as

$$ \bar{H} = L(x, u, w) + \frac{1}{2\alpha} \sum\limits_{i = 1}^{n_{d}}(d_{i}(x, u, w) + \nu_{i})^{2} + \lambda^{T}f(x, u, w) - \mu^{T} \nu + \frac{\alpha}{2}u(t)^{T}u(t), $$

where α > 0 is a penalty parameter, $\nu (t) \in \mathbb {R}^{n_d}$ are the non-negative slack variables, $\lambda (t) \in \mathbb {R}^{n_{x}}$ are the costate variables, $\mu (t) \in \mathbb {R}^{n_{d}}$ are the Lagrange multipliers associated with the non-negative slack variables ν(t).

If (x,u,w) represents a local minimum of the OCP (1)–(5), then it is necessary that the following conditions be satisfied.

$$ \begin{array}{@{}rcl@{}} \dot{x} &=& \frac{\partial \bar{H}}{\partial\lambda}, \end{array} $$

(58)

$$ \begin{array}{@{}rcl@{}} \dot{\lambda} &=& -\frac{\partial \bar{H}}{\partial x}, \end{array} $$

(59)

$$ \begin{array}{@{}rcl@{}} \dot{\gamma} &=& \frac{\partial \bar{H}}{\partial w}, \end{array} $$

(60)

$$ \begin{array}{@{}rcl@{}} 0 &=& \frac{\partial \bar{H}}{\partial u}, \end{array} $$

(61)

$$ \begin{array}{@{}rcl@{}} 0 &=& d_i + \nu_{i} - \alpha \mu_{i},\ i = 1,2,\ldots,n_d, \end{array} $$

(62)

$$ \begin{array}{@{}rcl@{}} 0 &=& \psi(\mu_{i}, \nu_{i}; \alpha),\ i=1,2,\ldots,n_d, \end{array} $$

(63)

$$ \begin{array}{@{}rcl@{}} 0 &=& {{{\varGamma}}}(x(t_{\mathrm{i}}),w), \end{array} $$

(64)

$$ \begin{array}{@{}rcl@{}} 0 &=& \lambda(t_{\mathrm{i}}) + D_{x}{{{\varGamma}}}(x(t_{\mathrm{i}}),w) \kappa_{\mathrm{i}}, \end{array} $$

(65)

$$ \begin{array}{@{}rcl@{}} 0 &=& \gamma(t_{\mathrm{i}}) - D_{w}{{{\varGamma}}}(x(t_{\mathrm{i}}),w) \kappa_{\mathrm{i}}, \end{array} $$

(66)

$$ \begin{array}{@{}rcl@{}} 0 &=& {{\varPsi}}(x(t_{\mathrm{f}}),w), \end{array} $$

(67)

$$ \begin{array}{@{}rcl@{}} 0 &=& \lambda(t_{\mathrm{f}}) - \frac{\partial\phi}{\partial x} - D_{x}{{\varPsi}}(x(t_{\mathrm{f}}),w) \kappa_{\mathrm{f}}, \end{array} $$

(68)

$$ \begin{array}{@{}rcl@{}} 0 &=& \gamma(t_{\mathrm{f}}) + \frac{\partial\phi}{\partial w} + D_{w}{{\varPsi}}(x(t_{\mathrm{f}}),w) \kappa_{\mathrm{f}}. \end{array} $$

(69)

where $\psi (\mu _{i}, \nu _{i}; \alpha ) = \mu _{i} + \nu _{i} - \sqrt {\mu _{i}^{2} + \nu _{i}^{2} + 2\alpha }$ is the Kanzow’s smoothed Fisher-Burmeister formula used.

The variables $\gamma (t) \in \mathbb {R}^{n_{w}}$ are used to write the stationary conditions associated with the parameters w as a differential equation instead of an integral equation. The Lagrange multipliers $\kappa _{\mathrm {i}} \in \mathbb {R}^{n_{{{\varGamma }}}}$ are associated with the initial time constraints and the Lagrange multipliers $\kappa _{\mathrm {f}} \in \mathbb {R}^{n_{{{\varPsi }}}}$ are associated with the final time constraints.

1.1.1 A.1.1 Boundary value problem

The necessary conditions (58)–(69) can then be represented in a compact form as a boundary value problem involving differential algebraic equations (BVP-DAEs) by separating the unknowns into: (i) differential variables; (ii) algebraic variables; and (iii) parameters. In particular, define

$$ y(t) = \left[\begin{array}{c} x(t) \\ \lambda(t) \\ \gamma(t) \end{array}\right],\ z(t) = \left[\begin{array}{c} u(t) \\ \mu(t) \\ \nu(t) \end{array}\right],\ p = \left[\begin{array}{c} w \\ K_{\mathrm{i}} \\ K_{\mathrm{f}} \end{array}\right], $$

(70)

where $y(t) \in \mathbb {R}^{n_{y}}$ are the differential variables, $z(t) \in \mathbb {R}^{n_{z}}$ are the algebraic variables, and $p \in \mathbb {R}^{n_{p}}$ are the parameter variables with n_y = 2n_x + n_w, n_z = n_u + 2n_d, and n_p = n_w + n_Γ + n_Ψ.

Using these definitions, the necessary conditions for a minimum can be written as the BVP-DAEs

$$ \begin{array}{@{}rcl@{}} \dot{y} &=& h(y(t),z(t),p), \end{array} $$

(71)

$$ \begin{array}{@{}rcl@{}} 0 &=& g(y(t),z(t),p,\alpha), \end{array} $$

(72)

$$ \begin{array}{@{}rcl@{}} 0 &=& r(y(t_{\mathrm{i}}),y(t_{\mathrm{f}}),p) \end{array} $$

(73)

Here, (71) defines a set of differential equations, (72) defines a set of algebraic equations, and (73) defines a set of boundary conditions. Moreover,

$$ h(y(t),z(t),p) = \left[\begin{array}{c} \partial \bar{H} / \partial \lambda \\ - \partial \bar{H} / \partial x \\ \partial \bar{H} / \partial w \end{array}\right] \in \mathbb{R}^{n_y}, $$

(74)

$$ g(y(t),z(t),p,\alpha) = \left[\begin{array}{c} \partial \bar{H} / \partial u \\ d(x,u,w) + Ne - \alpha Me \\ \psi_{d}(\mu,\nu;\alpha) \end{array}\right] \in \mathbb{R}^{n_z}, $$

(75)

$$ r(y(t_{\mathrm{i}}),y(t_{\mathrm{f}}),p) = \left[\begin{array}{c} {{{\varGamma}}}(x(t_{\mathrm{i}}),w) \\ \lambda(t_{\mathrm{i}}) + D_x{{{\varGamma}}}(x(t_{\mathrm{i}}),w) \kappa_{\mathrm{i}} \\ \gamma(t_{\mathrm{i}}) - D_{w}{{{\varGamma}}}(x(t_{\mathrm{i}}),w) \kappa_{\mathrm{i}} \\ {{\varPsi}}(x(t_{\mathrm{f}}),w) \\ \lambda(t_{\mathrm{f}}) - \frac{\partial\phi}{\partial x} - D_x{{\varPsi}}(x(t_{\mathrm{f}}),w) \kappa_{\mathrm{f}} \\ \gamma(t_{\mathrm{f}}) + \frac{\partial\phi}{\partial w} + D_{w}{{\varPsi}}(x(t_{\mathrm{f}}),w) \kappa_{\mathrm{f}} \end{array}\right] \in \mathbb{R}^{n_{y}+n_{p}}, $$

(76)

with $d(x,u,w) = [d_{1}(x,u,w), d_{2}(x,u,w), \ldots , d_{n_{d}}(x,u,w)]^{T}$, $\mathcal {N} = \text {diag} (\nu _{1},$ $\nu _{2}, \ldots ,\nu _{n_{d}})$, ${\mathscr{M}} = \text {diag} (\mu _{1}, \mu _{2}, \ldots ,\mu _{n_{d}})$, ψ_d(μ,ν; α) = [ψ(μ₁,ν₁; α),ψ(μ₂, $\nu _{2};\alpha ), \ldots , \psi (\mu _{n_{d}},\nu _{n_{d}};\alpha )]^{T}$, $D_x{{{\varGamma }}} = [\partial {{{\varGamma }}}_{1}/\partial x, \partial {{{\varGamma }}}_{2}/\partial x, \ldots ,$ $\partial {{{\varGamma }}}_{n_{{{{\varGamma }}}}}/\partial x]^{T}$, $D_{w}{{{\varGamma }}} = [\partial {{{\varGamma }}}_{1}/\partial w, \partial {{{\varGamma }}}_{2}/\partial w, \ldots , \partial {{{\varGamma }}}_{n_{{{{\varGamma }}}}}/\partial w]^{T}$, $D_x{{\varPsi }} = [ \partial {{\varPsi }}_{1}/\partial x, \partial {{\varPsi }}_{2}/\partial x, \ldots , \partial {{\varPsi }}_{n_{{{\varPsi }}}}/\partial x]^{T}$, $D_{w}{{\varPsi }} = [\partial {{\varPsi }}_{1}/\partial w, \partial {{\varPsi }}_{2}/\partial w, \ldots , \partial {{\varPsi }}_{n_{{{\varPsi }}}}/\partial w]^{T}$, and e = [1, 1,…, 1]^T.