Optimal Control of Differential Inclusions

Optimal control of differential inclusions explores dynamical systems governed by multivalued mappings, where the state derivative may assume values within a set. This field extends classical control theory by accommodating uncertainties, nonsmooth dynamics and discontinuities. Techniques draw upon viability theory, measure-theoretic selections and generalized differentiation to establish existence of optimal trajectories and derive necessary and sufficient conditions. Central tools include the Euler–Lagrange inclusion and Hamiltonian inclusion formulations, alongside a set-valued version of the Pontryagin maximum principle. Applications span robotics, where switching behaviour and path constraints arise; economics, modelling systems with regime changes; and engineering contexts such as traffic flow and population dynamics. Recent advances have emphasised numerical approximation schemes, discrete-time analogues and the treatment of state and control constraints within a unified framework, supporting both theoretical rigour and practical algorithmic implementation.

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Technical terms

Differential inclusion: A generalisation of a differential equation in which the derivative lies within a set-valued mapping rather than being single-valued.

Set-valued mapping: A rule assigning to each point in its domain a set of possible values rather than a single value.

Pontryagin maximum principle: A collection of necessary conditions characterising optimal trajectories and controls, extended to include multivalued dynamics.

Euler–Lagrange inclusion: A variational condition yielding necessary or sufficient optimality relations in the presence of multivalued dynamics.

Hamiltonian inclusion: A formulation of the Hamiltonian system adapted for differential inclusions, incorporating adjoint variables and subdifferentials.

References

1. Optimal control of hyperbolic type discrete and differential inclusions described by the Laplace operator. ESAIM Control Optimisation and Calculus of Variations (2022).
2. Duality in the problems of optimal control described by Darboux-type differential inclusions. Optimization Letters (2024).
3. Averaged optimal control problems of non-linear differential inclusions on the finite and infinite intervals. Науковий вісник Ужгородського університету Серія Математика і інформатика (2024).