Chemotherapy as a cancer treatment has garnered much scientific interest in recent years. Chemotherapy is essential for the elimination of cancer cells to the greatest extent possible. Clinical studies demonstrate a predictable trajectory of tumor volume after chemotherapy. However, the standard dose regimens often utilized in chemotherapy have substantial toxicity and little therapeutic value. Consequently, optimum drug dosage is crucial for effectively reducing the tumor size to a defined level while tracking a predefined course. This work establishes optimum control theory with L1-norm based cost function to discover the inputs that decrease the difference between real tumor growth and goal size to improve treatment efficacy while minimizing the side effects. Because of the bang-off nature of the L1-norm control profile, intermittent drug dosing is possible. This study also compares and contrasts the findings with the cost function based on L2-norm. Simulation findings validate the performance of the proposed control scheme for chemotherapy. Further study is needed to confirm the procedures in clinical situations and optimize the regimens for individual patients.

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In this paper, we have addressed a ROS (Reduced Order Synchronization) problem of two general classes of chaotic or hyperchaotic systems in master (drive) and slave (response) configuration. The master and the slave systems are considered to follow a very specific form of structure; so-called the strict-feedback form. Being a ROS problem, the order of the master system is naturally greater than that of the slave system. But we have taken a particular case where the order difference between both the systems is one. A systematic nonlinear back-stepping control design methodology is developed so to attain the reduced-order synchronization goal. In real practice, only [[EQUATION]] number of states of the response system get synchronized with [[EQUATION]] states of the drive system, incorporating [[EQUATION]] amount of controllers, having considered the order of the drive and response system to be [[EQUATION]] and [[EQUATION]] (and [[EQUATION]] ), respectively. It is shown that, unlike other existing conventional nonlinear control techniques, the back-stepping method requires only a single scalar controller [[EQUATION]] [[EQUATION]][[EQUATION]] to achieve the same task mentioned above. The question arises, what will happen to the [[EQUATION]] state of the master system. Being a chaotic system, it can be said that [[EQUATION]] state will be bounded in nature. But, its stability can not be guaranteed. By appending an extra [[EQUATION]] dynamics to the slave system and adding a suitable nonlinear active controller [[EQUATION]] [[EQUATION]] can tackle the stability challenge of the [[EQUATION]] state, as stated earlier. A sound generalization of the control function [[EQUATION]] and [[EQUATION]] have also been included in the discussion. To prove the asymptotic stability of the error dynamics of the system, the Lyapunov stability direct method has been utilized. Extensive MATLAB simulation results verify the theoretical findings.