## 1 Introduction  
Dynamic supply chains present novel approaches from mathematical modeling, control and stability analysis approaches. In the context of mathematical modeling, the use of ordinary differential equations (ODE), difference equations and partial differential equations (PDE) are suitable for model-based, control-oriented approaches. Several applications have been reported in dynamic supply chains: in [1], they applied optimal control to dynamic food supply chains; in [2], they created a dynamic optimization model by applying a real option on stochastic demands; and in [3], a fuzzy hybrid negotiation mechanism for dynamic SC is present, as well as in [4]. The purpose of this article is to consolidate the existing issues related to information systems in dynamic SC. In this research work, our aim is to focus on model-based optimal control theory. In terms of stability analysis, our goal is to analyze the stability of the dynamic SC by the application of linear and nonlinear systems. In operations and production-planning processes, decision-makers develop convenient and suitable decisions to validate their hypothesis about which decisions incorporate more profit while reducing costs on enterprise operations. In this way, a dynamic SC requires proper decision-making with effective performance and synchronization. In general, an SC processes raw material, cash, and information flows, taking into account the demand profile of the system. Manufacturing converts raw materials into end products within a demand profile. However, these manufacturing systems face problems, such as: long downtime, labor inefficiency, inaccurate scheduling, and weak energy consumption [5]. Recently, supply chain (SC) market forecasts in profitability have been increasing. Based on this, nowadays, SC has gained special attention among industry practitioners of automation science and control systems. SC applications in industry can integrate potential solutions, such as automotive, aerospace, biotechnology, energy, etc. Considering that manufacturing systems are dynamic and complex [6], the proper design of control systems requires flexibility and should be adaptive to process disturbances. In this context, new trends are required to take into account systems theory and control theory perspectives in order to support accurate, agile, and flexible solutions for SC applications. Industrial production systems require competitive and adaptive solutions in order to ensure profitability in markets. In manufacturing systems, decision-makers are required to achieve and approximate online decisions. A manufacturing SC is a system composed of suppliers, manufacturers, distributors, and customers that serves customer requirements [7]. SC integrates solutions to decrease operation management costs such as production, logistics, and quality. In this context, dynamic approaches for production-inventory systems are valuable in order to maximize profits and minimize costs throughout the SC system. Recently in SC, the focus has moved from the factory-level management to a more enterprise-level [8]. Inventory management presents a crucial role through the performance analysis of a supply chain. By definition, inventory management is \"the continuing process of planning, organizing, and controlling inventory while balancing supply and demand\" [9]. In this research work, our main interest is to provide modeling, analysis, and model-based optimal control for a dynamic SC from a systems theory perspective. Designing an inventory management system is directly related to sales, finance, production, and procurement [10]. Moreover, the goal of inventory management is to minimize the average cost per unit of time experienced for the inventory system in the long run [11]. Considering that decision variables are related to inventory levels along the SC, and production rate and capacity levels are fixed parameters, this paper presents the mixing problem as an analogy for an approximation of a four-echelon supply chain.