Ashutosh Bijalwan defended his PhD thesis Numerical optimisation of worm locomotion on frictional substrates, supervised by José J. Muñoz on 18th July 2024 within the UPC doctoral program in Applied Mathematics. Currently he is in ExxonMobil working as a research engineer in India.
Thesis summary
Optimal control problems may be posed as the minimisation of a functional subject to ODE or PDE constraints. The solution of such problems is very relevant from the practical perspective due to the wide type of applications where they may be found: trajectory planning, autonomous vehicles, or shape analysis [1]. The optimality conditions of the state and control variables has been historically controversial, but the prevalent strategy is to resort to the so-called Pontryagin’s maximum principle [2]. The resulting set of differential equations for the state, adjoint and control variables share many notable properties with Hamiltonian systems: they have underlying integrals of motion and (pre-)symplectic structures, which interestingly prevail even in the presence of external forces or dissipation effects.
The numerical solution of the optimality conditions is not less challenging, since they combine initial and final value problems, together with algebraic equations. In this thesis we have proposed numerical schemes that preserve as many as the properties of the analytical solution, such as the invariance of the Hamiltonian or the preservation of the flow in the phase-space of state and adjoint variables [3], while garanteeing the stability of the numerical scheme [6].
The thesis adapts the time-integration schemes and solution strategies to problems in continuum elastodynamics and multibody systems [5], and applies them to inverted elastic pendulum or to find the optimal contractility profiles of worm-like bodies [4], as shown in Fig. 3. A salient result of the thesis is that anisotropic friction is necessary for the net motion of the worm, a condition that parallels the celebrated scallop theorem for microswimmers in high Reynolds numbers 7.
Bijalwan_Fig1
Figure 3: Optimal contractility patterns and corresponding deformations for different worm locomotion strategies. Each row is a different strategy (inching, crowling undulatory), and each column a different time step.
References
[1] MI Miller and A Trouvé and Laurent Younes. Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D’Arcy Thompson. Annu. Rev. Biomed. Eng. 2015. 17:447–509
[2] H Pesch and M Plail. The cold war and the maximum principle of optimal control. Optimization Stories. Documenta Mathematica (2012)
[3] A Bijalwan and JJ Muñoz. A control Hamiltonian preserving discretisation for optimal control. Multibody System Dynamics, Vol. 59, pp. 19-43, 2023.
[4] A Bijalwan and JJ Muñoz. Adjoint-based optimal control of contractile elastic bodies. Application to limbless locomotion on frictional substrates. Comp. Meth. Appl. Mech. Eng., Vol. 420, 116697, 2024.
[5] A Bijalwan and S Schneider and P Betsch and JJ Mu\~noz. Monolithic and staggered solution strategies for constrained mechanical systems in optimal control problems. Under review.
[6] A Bijalwan, JJ Muñoz. “On the Numerical Stability of Discretised Optimal Control Problems”. Optimal Design and Control of Multibody Systems. Eds: Karin Nachbagauer, Alexander Held. IUTAM Bookseries, Vol 42, 2024.
[7] Purcell, E. M. «Life at low reynolds number», American Journal of Physics, 45 (1): 3–11, 1977.