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Runners competing in races are looking to optimize their performance. In this paper, a runner's performance in a race, such as a marathon, is formulated as an optimal control problem where the controls are: the nutrition intake throughout the race and the propulsion force of the runner. As nutrition is an integral part of successfully running long distance races, it needs to be included in models of running strategies.

We formulate a system of ordinary differential equations to represent the velocity, fat energy, glycogen energy, and nutrition for a runner competing in a long-distance race. The energy compartments represent the energy sources available in the runner's body. We allocate the energy source from which the runner draws, based on how fast the runner is moving. The food consumed during the race is a source term for the nutrition differential equation. With our model, we are investigating strategies to manage the nutrition and propulsion force in order to minimize the running time in a fixed distance race. This requires the solution of a nontrivial singular control problem.

As the goal of an optimal control model is to determine the optimal strategy, comparing our results against real data presents a challenge; however, in comparing our results to the world record for the marathon, our results differed by 0.4%, 31 seconds. Per each additional gel consumed, the runner is able to run 0.5 to 0.7 kilometers further in the same amount of time, resulting in a 7.75% increase in taking five 100 calorie gels vs no nutrition.

Our results confirm the belief that the most effective way to run a race is to run approximately the same pace the entire race without letting one's energies hit zero, by consuming in-race nutrition. While this model does not take all factors into account, we consider it a building block for future models, considering our novel energy representation, and in-race nutrition.

 

For polyhedral constrained optimization problems and a feasible point$$\textbf{x}$$$x$, it is shown that the projection of the negative gradient on the tangent cone, denoted$$\nabla _\varOmega f(\textbf{x})$$${\nabla }_{\Omega }f\left(x\right)$, has an orthogonal decomposition of the form$$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$$\beta \left(x\right)+\phi \left(x\right)$. At a stationary point,$$\nabla _\varOmega f(\textbf{x}) = \textbf{0}$$${\nabla }_{\Omega }f\left(x\right)=0$so$$\Vert \nabla _\varOmega f(\textbf{x})\Vert $$$\Vert {\nabla }_{\Omega }f\left(x\right)\Vert$reflects the distance to a stationary point. Away from a stationary point,$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$measure different aspects of optimality since$$\varvec{\beta }(\textbf{x})$$$\beta \left(x\right)$only vanishes when the KKT multipliers at$$\textbf{x}$$$x$have the correct sign, while$$\varvec{\varphi }(\textbf{x})$$$\phi \left(x\right)$only vanishes when$$\textbf{x}$$$x$is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$.

 

A mesh refinement method is described for solving optimal control problems using Legendre‐Gauss‐Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targetedh‐refinement approach whereby the discontinuity locations are bracketed with mesh points. The remaining smooth portions of the mesh are refined using previously developed techniques. The method is demonstrated on two examples, and results indicate that the method solves optimal control problems with discontinuous control solutions using fewer mesh refinement iterations and less computation time when compared with previously developed methods.

 

In an earlier paper (https://doi.org/10.1137/21M1393315), the switch point algorithm was developed for solving optimal control problems whose solutions are either singular or bang-bang or both singular and bang-bang, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. The class of control problems that were considered had a given initial condition, but no terminal constraint. The theory is now extended to include problems with both initial and terminal constraints, a structure that often arises in boundary-value problems. Substantial changes to the theory are needed to handle this more general setting. Nonetheless, the derivative of the cost with respect to a switch point is again the jump in the Hamiltonian at the switch point. 

The Polyhedral Active Set Algorithm (PASA) is designed to optimize a general nonlinear function over a polyhedron. Phase one of the algorithm is a nonmonotone gradient projection algorithm, while phase two is an active set algorithm that explores faces of the constraint polyhedron. A gradient-based implementation is presented, where a projected version of the conjugate gradient algorithm is employed in phase two. Asymptotically, only phase two is performed. Comparisons are given with IPOPT using polyhedral-constrained problems from CUTEst and the Maros/Meszaros quadratic programming test set.

 

The trajectory optimization of a reusable launch vehicle entry is studied. The objective is to maximize the crossrange during atmospheric entry subject to a constraint on the stagnation point heating rate. The problem is solved by partitioning the domain of the independent variable into multiple subdomains such that each subdomain consists of a segment where the heating rate constraint is either active or inactive. Additional necessary conditions for optimality are enforced in segments where the heating rate constraint is active. A multiple-domain Legendre-Gauss-Radau direct collocation method is then used to solve the partitioned problem. Key features of the stagnation point heating rate constraint are presented, and it is observed that the aforementioned approach is capable of solving the problem under consideration more accurately than traditional direct collocation methods. 

A Monte Carlo analysis of a contingency optimal guidance strategy is conducted. The guidance strategy is applied to a Mars Entry problem in which it is assumed that the surface level atmospheric density is a random variable. First, a nominal guidance strategy is employed such that the optimal control problem is re-solved at constant guidance cycles. When the trajectory lies within a particular distance from a path constraint boundary, the nominal guidance strategy is replaced with a contingency guidance strategy, where the contingency guidance strategy attempts to prevent a violation in the the relevant path constraint. The contingency guidance strategy utilizes the reference optimal control problem formulation, but modifies the objective functional to maximize the margin between the path constraint limit and path constraint function value. The ability of the contingency guidance strategy to prevent violations in the path constraints is assessed via a Monte Carlo simulation. 

A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc. 

A Monte Carlo analysis of a contingency optimal guidance strategy is conducted. The guidance strategy is applied to a Mars Entry problem in which it is assumed that the surface level atmospheric density is a random variable. First, a nominal guidance strategy is employed such that the optimal control problem is re-solved at constant guidance cycles. When the trajectory lies within a particular distance from a path constraint boundary, the nominal guidance strategy is replaced with a contingency guidance strategy, where the contingency guidance strategy attempts to prevent a violation in the the relevant path constraint. The contingency guidance strategy utilizes the reference optimal control problem formulation, but modifies the objective functional to maximize the margin between the path constraint limit and path constraint function value. The ability of the contingency guidance strat- egy to prevent violations in the path constraints is assessed via a Monte Carlo simulation.