## Inverse problems & control

The study of soft materials has extensively focused on the ‘forward problem’, i.e., predicting the macroscopic properties, patterns, and dynamics of a system given its composition, microscopic properties, interactions, etc. But in engineering and biological contexts, one is interested in asking how such systems can be controlled and made functional by manipulating material parameters, i.e., the *inverse problem*. Inverse problems are often associated with inference (given data/observation, what is the model?) or accomplishing functional goals/tasks, and as a result, they are often ill-posed, i.e., multiple solutions exist. What principles can we use to regularize such inverse problems? How can we design physically constrained control strategies to manipulate soft, active materials? We use ideas based on control theory, optimization, and symmetry to develop interpretable frameworks and explore some of these questions in the context of controlling active fluids. The control and inverse design of soft, active matter is a major focus of our current research.

One basic task is transportation. While much is known about optimal and efficient strategies to move matter, energy, and information around, how can we craft similar optimal protocols to transport autonomously moving (active) matter, such as self-propelled drops or migrating cells? We have developed an optimal control framework to transport drops of an active suspension with the least amount of energy dissipated, by manipulating the spatiotemporal profile of its internal active stresses. A natural ‘gather-move-spread’ strategy emerges as an optimal solution that trades-off size and shape variations with translation as a function of active and passive forces in the fluid drop. Our work suggests general principles for optimal transportation in a wide variety of synthetic and biological active systems.

Manipulating spatially extended continuous materials, such as bulk active fluids, requires different strategies. One approach is to focus on robust localized excitations, such as topological defects, as discrete building blocks to hierarchically structure the material. In active fluids, topological defects are often intimately coupled with flow, begging the question – how can we control the flow and dynamics of active defects by spatiotemporally manipulating active stresses? By using a symmetry-based, additive control framework, we have obtained design rules to construct ‘active topological tweezers’ that can transport and manipulate defects along complex space-time trajectories. Similar principles apply for other complex patterning tasks, including at the collective level.

#### Relevant publications

**Spatiotemporal control of active topological defects**, S. Shankar, L. V. Scharrer, M. J. Bowick, M. C. Marchetti, 2022 [arxiv:2212.00666]**Optimal transport and control of active drops**, S. Shankar, V. Raju, L. Mahadevan, PNAS 119 (35) e2121985119 (2022) [DOI] [PDF] [arXiv]**Hydrodynamics of Active Defects: from order to chaos to defect ordering**, S. Shankar, M. C. Marchetti, Phys. Rev. X 9, 041047 (2019) [DOI] [PDF] [arXiv]**Boundaries control active channel flows**, P. Gulati, S. Shankar, M. C. Marchetti, Front. Phys. 10:948415 (2022)

[DOI] [PDF] [arXiv]