We study the optimal consumption problem in the neoclassical theory of capital accumulation and economic growth under uncertainty. We consider two factor markets – one for labor and one for capital services.

There are many identical households, each with a utility function given by U(c) for consumption c. Households supply the uncertain amount of labor Y(t). There are many identical firms, each with the same technology for production. Firms rent the services of capital and labor to produce output. The increase of the capital stock X(t) coincides with the totality of the production F(X(t), Y(t)) net of the consumption c(t) and the depreciation λX(t) with rate λ until the capital stock vanishes. The decision that the household has to make is how much to consume or how to maximize the expected discounted utility of consumption with a utility function U(c).

We show the existence of a classical solution of the HJB equation (9.6) associated with the stochastic optimization problem, and then give an optimal consumption policy in terms of its solution.

The Model

Consider the neoclassical growth model of the Solow-type under uncertainty, in Merton [113]. Define the following quantities:

1. Y(t) = labor supply at time t.

2. X(t) = capital stock at time t.

3. F(x, y) = constant-returns-to-scale production function producing the commodity for the capital stock x ≥ 0 and the labor force y > 0.

4. […]

In this chapter, we study the theory of optimal stopping problems in mathematical finance. We consider an investor, that is, an economic agent who can trade continuously in the stock market.

The stock price X(t) is governed by a stochastic differential equation with an expected return r. An American call or put option gives the right to buy or to sell the underlying asset with the reward function g(X(t)) at any time t. The objective of the investor is to decide the exercise time τ when he buys or sells the risky asset for the maximal expected reward E[e-rτg(X(τ))].

By the method of penalization, we solve the variational inequality associated with this problem. The value function coincides with its solution v and, by using v, the optimal stopping time is shown to exist.

The Model

Consider the optimal stopping problem for the stock price in mathematical finance. Define the following quantities:

1. X(t) = stock price at time t.

2. r = expected return of the stock, r > 0.

3. B(t) = the standard Brownian motion.

4. σ = the nonzero diffusion constant.

5. τ = exercise time or stopping time.

6. g(x) = reward function of stock x.

We assume that the stock price X = {X(t)} evolves according to the stochastic differential equation,

on a complete probability space (Ω,ℱ,P), carrying a standard Brownian motion {B(t)}, endowed with the natural filtration ℱt generated by σ(B(s), s ≤ t).

This chapter is concerned with the consumption and portfolio selection problems formulated by R. C. Merton [112] in mathematical finance. We consider the optimization problems to maximize the expected utility function with respect to consumption rates c(t) and portfolio processes π(t) for the vector process S(t) of prices of N risky assets S1(t), …, SN(t) and general utility functions U(c). We try to find a connection between the HJB equations and the linear differential equations by using the Legendre transform. We study the existence of the smooth solutions of the HJB equations from the point of view of viscosity solutions.

The Model

Consider the consumption and portfolio selection problem. Define the following quantities:

1. S0(t) = price of the riskless asset at time t.

2. S(t) = vector process of prices of the N risky assets S1(t), …, SN(t) at time t.

3. B(t) = the N-dimensional standard Brownian motion.

4. X(t) = total wealth at time t.

5. π(t) = fraction of wealth in the risky assets at time t.

6. r = return on the riskless asset.

7. b = vector of expected return on the N risky assets.

8. σ = volatility N × N matrix of risky assets, σ ≠ 0.

9. c(t) = consumption rate at time t.

10. U(c) = the utility function.

Mixed logical dynamical systems and linear complementarity systems are representations of switched systems, which under the conditions described here are equivalent to the model used in Chapter 4. They are particularly useful for model-predictive control. The equivalences of several hybrid system models show that different models, which are suitable for specific analysis and design problems and have been investigated in detail, cover the same class of hybrid systems. The analysis of the well-posedness of the models leads to conditions on the model equations under which a unique solution exists.

Model-predictive control of hybrid systems

Model-predictive control (MPC) is a widely used technology in industry for control design of highly complex multivariable processes. The idea behind MPC is to start with a model of the open-loop process that explains the dynamical relations among system's variables (command inputs, internal states, and measured outputs). Then, constraint specifications on system variables are added, such as input limitations (typically due to actuator saturation) and desired ranges where states and outputs should remain. Desired performance specifications complete the control problem setup and are expressed through different weights on tracking errors and actuator efforts (as in classical linear quadratic regulation). At each sampling time, an open-loop optimal control problem based on the given model, constraints, weights, and with initial condition set at the current (measured or estimated) state, is repeatedly solved through numerical optimization.

By considering a solar air conditioning plant, the typical steps for analyzing and controlling hybrid systems under practical circumstances are described in this chapter.

Plant description

The present chapter describes the application and the implementation of a hybrid control scheme of a solar air conditioning plant. The conditioning plant considered is a hybrid system characterized by a variable configuration, with discrete and continuous variables, and components that change their dynamics according to the conditions under which the plant operates. Section 17.1 describes the solar air conditioning plant. Section 17.2 shows briefly the hybrid modeling of the plant. Section 17.3.1 describes the control requirements to operate the plant. Section 17.3.2 develops a hybrid control strategies for the operation of the plant. Section 17.3.3 shows the experimental results and discusses them.

Main components

The solar air conditioning plant considered in this chapter is located in Seville (Spain) and is used to cool down the laboratories of the Department of System Engineering and Automation of the University of Seville. It consists of a solar field producing hot water that feeds into an absorption machine, generating chilled water and injecting it into the air distribution system, which has a cooling power of 35 kW.

Figure 17.1 offers a general scheme of the plant, and shows its main components: the solar subsystem, composed of a set of flat solar collectors; the accumulation subsystem, composed of two tanks storing hot water; and the cooling machine.

Hybrid systems are dynamical systems that consist of components with continuous and discrete behavior. Modeling, analysis, and design of such systems raise severe methodological questions, because they necessitate the combination of continuous variable system descriptions like differential and difference equations with discrete-event models like automata or Petri nets. Consequently, hybrid systems methodology is based on the principles and results of the theories of continuous and discrete systems, which, until recently, have been elaborated separately, with contributions coming from different disciplines, such as control theory, computer science, and mathematics.

This handbook reviews the new phenomena and theoretical problems brought about by the combination of continuous and discrete dynamics and surveys the main approaches, methods, and results that have been obtained during the last decade of research in this field. It is structured into three main parts:

• Part I: Modeling, analysis, and control design methods: The first part gives a thorough introduction to hybrid systems theory. The material is classified by the modeling approaches used to represent hybrid systems in a form that is convenient for analysis and control design. Hybrid automata and switched systems are well-studied system classes, which are extensively described, but other approaches like mixed logical dynamical systems, complementarity systems, quantized systems, and stochastic hybrid systems are also explained.

• Part II: Tools: The second part is concerned with computer-aided systems analysis, control design, and verification. After a survey of the variety of relevant tools, selected tools are described in more detail. […]

An overview of various modeling frameworks for hybrid systems is given followed by a comparison of the modeling power and the model complexity, which can serve as a guideline for choosing the right model for a given analysis or control problem with hybrid dynamics. Then, the main analysis and design tasks for hybrid systems are surveyed together with the methods for their solution, which will be discussed in more detail in subsequent chapters.

Models for hybrid systems

Overview

As models are the ultimate tools for obtaining and dealing with knowledge, not only in engineering, but also in philosophy, biology, sociology, and economics, a search has been undertaken for appropriate mathematical models for hybrid systems. This section gives an overview of the modeling formalisms that have been elaborated in hybrid systems theory in the past.

Structure of hybrid systems Many different models have been proposed in literature, as will be seen in following chapters. These models can be distinguished with respect to the phenomena that they are able to represent in an explicit form. Consequently, these models have different fields of applications. The main idea of these models is described by the block diagram shown in Fig. 2.1, which is often used in literature as a starting point of hybrid systems modeling and analysis, although not all models use this structure in a direct way.

Control loops which are closed over a digital communication network became a topic of intensive research in the recent years. This chapter surveys the main problems to be solved and show how hybrid systems theory can help to solve them. The chapter ends with a case study that was inspired by a practical application of hybrid systems methods in an ore mine.

Introduction to distributed control applications and networked control systems

In this chapter, we deal with control over networks, i.e. with control implementations where the control actions and decisions are taken based on measurement, decisions, and actuations that take place in a distributed environment. The control agents may rely upon a centralized facility that coordinates and optimizes the overall control strategy and on a shared communication resource like a bus (distributed control) or may be acting only on local information and on data exchanged with neighboring nodes (decentralized control). There are obviously pros and cons for each strategy. Decentralized control systems have the following characteristics:

• There is no central control node.

• There is no common communication facility; communication is point-to-point.

• The global network topology is unknown to the nodes, which are only aware of their neighborhood.

These features yield interesting properties:

• The system is scalable: there are no limits imposed by centralized computing power or global communication bandwidth.

• The system is robust and fault tolerant, because it supports dynamical changes of the network topology and losses of nodes. […]

Hybrid automata is a modeling formalism for hybrid systems that results from an extension of finite-state machines by associating with each discrete state a continuous-state model. Conditions on the continuous evolution of the system invoke discrete state transitions. A broad set of analysis methods is available for hybrid automata including methods for the reachability analysis, stability analysis, and optimal control.

Definition

A hybrid automaton is a transition system that is extended with continuous dynamics. It consists of locations, transitions, invariants, guards, n-dimensional continuous functions, jump functions, and synchronization labels. Various definitions exist in the literature which differ only in details. The following definition covers all of the aspects needed for the purpose of this handbook. A hybrid automaton consists of:

• a finite set of locations Q, q ∈ Q;

• a finite transition relation ⊖ ⊆ Q × Q

for the specification of the discrete dynamics. The locations can be seen as discrete states (also called control modes), in other words as the discrete part of the hybrid state space ℋ. Transitions from one control mode to the next are often called control switches. The continuous dynamics is described by:

• a finite and indexed set of continuous variables V = {x1, x2, …, xn}, often written as a vector x = (x1, …, xn);

• a real-valued activity function f : Q × ℝn → ℝn, often defined by a continuous differential equation ẋ = dx/dt = f(q, x).