Complex dynamical systems exist in many theoretical and practical domains of science and engineering, including physical processes, man-made systems, networks of leader-follower multi-agent systems, and distributed deterministic and stochastic control systems. The matrix approach to state space has long been considered the optimal way of addressing many of the central problems of control systems. In recent decades, novel methods and approaches to the study of these systems, both linear and nonlinear, have been based on a geometrical approach in which the objective is to reveal the properties of the geometric skeleton of the dynamic system. The geometric approach can convert a difficult nonlinear problem into a straightforward linear one. Geometric control theory and sub-Riemannian Geometry are domains that play an important role in complex dynamical systems. Geometric control theory, considered as a unification of the geometric theory of differential systems with symplectic geometry, and the Maximum Principle, has the same common field with differential geometry and mechanics. Smooth dynamical systems are governed by a family of ordinary differential equations, parametrized by control parameters. Applying Lie theory techniques, geometric control theory allows designing controllers for the orientation of robots and satellites using geometric mechanics.

Sub-Riemannian Geometry deals with spaces whose metric structure provides a constrained geometry. It can be applied in the study of optimal control and path planning, where the space of motion of vehicles has limited degrees of freedom. This geometry is also relevant in a variety of physical and biological systems, such as the neurobiological functional mechanism of the first layer of the mammalian visual cortex, associated with intra-cortical communication, where we may model it using a sub-Riemannian structure. In digital image reconstruction, the role of sub-Riemannian geometry in completing missing or occluded image data from the retina is relevant, solving the minimal surface with Dirichlet boundary conditions. Control theory allows us to develop analytical tools to study the controllability and good dynamical response of complex systems emerging in nature and engineering, for instance, the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system’s entire dynamics.

Meanwhile, Polynomial theory has been a useful tool to explain the classical and complex behaviour of dynamical systems, for instance, considering the stability behaviour of a dynamical system, a computational search for parameter-dependent transitions can be affected by doing algebraic operations with the coefficients of the characteristic polynomial of the corresponding system. In addition, recently, polynomial approaches have been used to study the chaotic behaviour of complex systems, particularly to generate scrolls. The polynomial approaches have also been exploited to solve fundamental problems such as controllability, stability, and robustness. Polynomial’s theory can also be applied in uncertain, nonlinear, time-delay and hybrid systems and model predictive control.

The aim of this Special Issue is to develop theoretical and practical methods and tools, useful models, and differential-algebraic and geometric techniques for the analysis of problems in the domain of complex systems. Authors are encouraged to submit papers that discuss new directions in the way of original research in the field of complex systems. Review articles are also encouraged.

Potential topics include but are not limited to the following:

• Optimal control of complex systems
• Polynomial approaches for studying the stability of continuous complex systems
• Control of complex systems and sub-Riemannian geometry
• Polynomial approaches for studying the stability of discrete complex systems
• Control, modeling, and numerical estimation of complex systems
• Polynomial approaches for studying chaotic behavior of complex systems
• Engineering of complex systems and intelligent robotics design
• Sub-Riemannian metric complexity
• Matrix approach for complex systems
• Polynomial approach for control of complex behavior
• Pattern recognition in complex systems
• Analysis and control of complex networks