Mathematical System of Lorenz-Rössler Model: Optimal Control and Bifurcation Issues - Alwan, Saba M.; Al-Mahdi, Abdo M.; Odhah, Omalsad H. (Mathematics and Computer Science: Research Updates Vol. 2, 2025)
Optimal control is one of the most popular decision-making tools recently in many research and in many areas. An optimal control problem involves: A mathematical form for the system to be controlled, a description of the constraints, determination of the goal to be accomplished, usually it is an additional boundary condition and determination for the performance measure. The Lorenz-Rössler model is one of the interesting models because of the idea of consolidation of the two models Lorenz and Rössler. The Lorenz-Rössler system is a three-dimensional system with five parameters. This paper discusses the Lorenz-Rössler model from where the bifurcation phenomena and the optimal control problem (OCP). The bifurcation property at the system equilibrium (0,0, b2/b1) is studied and it is found that saddle-node and Hopf bifurcations can be holed under some conditions on the parameters. Also, the problem of the optimal control of the Lorenz-Rössler model is discussed and using Pontryagin’s Maximum Principle (PMP) to derive the optimal control inputs that achieve the optimal trajectory. Numerical examples and solutions for bifurcation cases and the optimal controlled system are carried out and shown graphically to show the effectiveness of the used procedure. Integrating bifurcation analysis and optimal control enhances our understanding of the dynamics and controllability of such chaotic systems. The manuscript is particularly valuable for advancing decision-making techniques in mathematical modelling and its applications across various domains, including secure communications and economic systems.