# Dynamic Optimization Using Local Collocation Methods and Improved Multiresolution Technique
## 1 Introduction  
Many scientific and industrial tasks, such as the optimal control of chemical processes and bioprocesses as well as control optimization in aerospace engineering, can be formulated as dynamic optimization problems (also named optimal control problems). Analytical solutions to dynamic optimization problems are difficult or even impossible to achieve. As a result, dynamic optimization problems are often solved using numerical methods. The most used numerical methods include direct methods [1; 2; 3] and indirect methods [4; 5; 6]. Indirect methods transform dynamic optimization problems into boundary value problems using the variational method or Pontryagin's minimum principle. The primary advantage of indirect methods is that they satisfy the necessary optimality conditions. However, the boundary value problems are rather difficult to solve in most cases because they are very sensitive to initial values of adjoint variables. Direct methods discretize the control variables or both the control and state variables, and transform dynamic optimization problems into nonlinear programming (NLP) problems. The resulting NLP problems can be solved by standard NLP solvers such as the sparse nonlinear optimizer (SNOPT) [7] and the interior point optimizer (IPOPT) [8]. Some recommended literature reviews on numerical methods for solving dynamic optimization problems can be found in [9; 10; 11; 12]. The use of direct methods to solve dynamic optimization problems has become more and more popular in recent years [3]. The major reason is that direct methods avoid the mathematical derivation of the necessary optimality conditions, which may be rather challenging for complex dynamic optimization problems. Furthermore, it is easy for direct methods to incorporate state and control constraints. Most of all, studies have reveal that direct methods are generally more robust withrespect to inaccurate initial guesses, and therefore converge more easily. Direct methods can be further classified as direct shooting methods (also named control vector parameterization), which discretize only the control variables in the time horizon, and direct collocation methods, which discretize not only the control variables but also the state variables. The primary advantage of direct shooting methods is that the resulting NLP has a small number of variables [3]. The disadvantage of direct shooting methods is that the solution is very sensitive to the initial condition [3]. This high sensitivity can result in very nonlinear NLP constraints that are therefore very difficult to solve. Additionally, the nonlinearity in NLP constraints makes it difficult to construct accurate estimates of the first- and second-order derivative matrixes required for gradient-based optimization algorithms. The high dependence of the solution on the initial condition of shooting methods can be alleviated by multiple shooting methods [3; 13; 14] that divide the time horizon into a number of subintervals and then perform the shooting methods in each of the subintervals. Additional constraints are needed for multiple shooting methods to ensure continuity of state variables between adjacent subintervals. The NLP resulting from direct collocation methods, on the other hand, has many more decision variables because the methods discretize both the state variables and the control variables. However, the NLP problems from direct collocation methods are more robust to initial guesses of the decision variables and thus are easier to converge [2; 15; 16]. In aerospace engineering [3; 12], both direct shooting methods and direct collocation methods have been widely used, and direct collocation methods are used even more often due to their excellent performance at solving complex dynamic optimization problems. In the optimization of chemical process or bioprocess control problems, direct shooting methods [14; 17; 18; 19; 20; 21; 22] appear to be used more often than direct collocation methods [3; 23; 24], especially when used in conjunction with mesh refinement methods.