This chapter, an updated and enlarged version of Chapter 4 from the 4th Edition of "Dynamic Programming and Optimal Control" by Dimitri P. Bertsekas, focuses on noncontractive total cost problems. These problems lack the contractive structure found in discounted problems, and the chapter aims to provide a unified treatment of several models, including positive and negative cost models, deterministic optimal control, stochastic shortest path models, and risk-sensitive models. Key points include: - **Stochastic Shortest Path Problems**: The chapter discusses weak conditions for these problems and their relation to positive cost problems. - **Deterministic Optimal Control**: It covers deterministic optimal control and adaptive dynamic programming. - **Affine Monotonic and Multiplicative Cost Models**: The chapter explores these models, which include multiplicative cost problems and their connection to stochastic shortest path problems. The chapter also delves into the analysis of these models, highlighting the importance of uniform positivity or negativity of the cost per stage, deterministic problem structures, and cost-free absorbing termination states. It presents results on the optimal cost function and optimal stationary policies, and discusses the convergence of value iteration (VI) and policy iteration (PI) algorithms under different assumptions. The chapter emphasizes the need for additional assumptions beyond the basic contractive structure to ensure the validity of these algorithms, and provides conditions for their convergence. It also introduces computational methods such as VI and PI, and discusses the limitations and challenges of these methods in noncontractive problems. Finally, the chapter includes a section on finite-state positive cost models, showing that these problems can be transformed into equivalent stochastic shortest path problems, allowing the application of the analysis and computational techniques from Chapter 3.This chapter, an updated and enlarged version of Chapter 4 from the 4th Edition of "Dynamic Programming and Optimal Control" by Dimitri P. Bertsekas, focuses on noncontractive total cost problems. These problems lack the contractive structure found in discounted problems, and the chapter aims to provide a unified treatment of several models, including positive and negative cost models, deterministic optimal control, stochastic shortest path models, and risk-sensitive models. Key points include: - **Stochastic Shortest Path Problems**: The chapter discusses weak conditions for these problems and their relation to positive cost problems. - **Deterministic Optimal Control**: It covers deterministic optimal control and adaptive dynamic programming. - **Affine Monotonic and Multiplicative Cost Models**: The chapter explores these models, which include multiplicative cost problems and their connection to stochastic shortest path problems. The chapter also delves into the analysis of these models, highlighting the importance of uniform positivity or negativity of the cost per stage, deterministic problem structures, and cost-free absorbing termination states. It presents results on the optimal cost function and optimal stationary policies, and discusses the convergence of value iteration (VI) and policy iteration (PI) algorithms under different assumptions. The chapter emphasizes the need for additional assumptions beyond the basic contractive structure to ensure the validity of these algorithms, and provides conditions for their convergence. It also introduces computational methods such as VI and PI, and discusses the limitations and challenges of these methods in noncontractive problems. Finally, the chapter includes a section on finite-state positive cost models, showing that these problems can be transformed into equivalent stochastic shortest path problems, allowing the application of the analysis and computational techniques from Chapter 3.