Caratheodory's Approach Research Articles

New approach to classical thermodynamics

The author constructs a new conception of thermodynamics which is based on new results in number theory. We consider a maximally wide range of gases, liquids, and fluids to which, in principle, the Caratheodory approach can be applied. The Caratheodory principle is studied using the Lennard-Jones potential as an example. On the basis of this example, we analyze the dispersive structure of a fluidwhose density exceeds the critical value. We introduce a new parameter, the “jamming factor,” which determines the jamming effect for such fluids. A comparison with experimental data for nonpolar molecules is carried out. The phase transition “liquid-amorphous solid” is studied in detail in the domain of negative pressures. We discuss the theoretical relationship between the obtained solutions and econophysics, some mysteries in biology, and other sciences.

An Outer-Measure Approach for Resource-Bounded Measure

Lutz developed a general theory of resource-bounded measurability and measure on suitable complexity classes C⊆C (see Proceedings of the 13th IEEE Conference on Computational Complexity, pp. 236–248, 1998), where Cantor Space C is the class of all decision problems, and classes C include various exponential time and space complexity classes, the class of all decidable languages, and the Cantor space C itself. In this paper, a different general theory of resource-bounded measurability and measure on those complexity classes is developed. Our approach is parallel to the Caratheodory outer measure approach in classical Lebesgue measure theory. We shall show that many nice properties in the classical Lebesgue measure theory hold in the resource-bounded case also. The Caratheodory approach gives short and easy proofs of theorems in the resource-bounded case as well as in the classical case. The class of measurable sets in our paper is strictly larger than that of Lutz, and the two resource-bounded measures assign the same measure for a set if the set is measurable in the sense of Lutz.

Notes on the third law of thermodynamics: II

We analyse some aspects of the third law of thermodynamics. We first review both the entropic version (N) and the unattainability version (U) and the relation occurring between them. Then, we heuristically interpret (N) as a continuity boundary condition for thermodynamics at the boundary T = 0 of the thermodynamic domain. On a rigorous mathematical footing, we discuss the third law both in Caratheodory's approach and in Gibbs' one. Caratheodory's approach is fundamental in order to understand the nature of the surface T = 0. In fact, in this approach, under suitable mathematical conditions, T = 0 appears as a leaf of the foliation of the thermodynamic manifold associated with the non-singular integrable Pfaffian form δQrev. Being a leaf, it cannot intersect any other leaf S = const of the foliation. We show that (N) is equivalent to the requirement that T = 0 is a leaf. In Gibbs' approach, the peculiar nature of T = 0 appears to be less evident because the existence of the entropy is a postulate; nevertheless, it is still possible to conclude that the lowest value of the entropy S has to be attained at the boundary of the convex set where S is defined.

Carnot Cycle Revisited

Sadi Carnot stated that the efficiency of a reversible Carnot cycle is independent of the properties of the material used to run the cycle. Using this statement, all textbook discussions of the Carnot cycle use an ideal gas. Here, in contrast, we consider, in the spirit of the Caratheodory approach, a general analysis centered on the existence of an integrating factor that transforms an inexact differential into an exact differential. Also we derive a general relation between temperature and volume along an adiabatic path. Using these two results, we obtain the efficiency of the Carnot cycle, η = 1 − TC/TH, for any equation of state.

The maximum principle, Bellman's equation, and Carathéodory's work

One of the most important and deep results in optimal control theory is the maximum principle attributed to Hestenes (1950) and in particular to Boltyanskii, Gamkrelidze, and Pontryagin (1956). Another prominent result is known as the Bellman equation, which is associated with Isaacs' and Bellman's work (later than 1951). However, precursors of both the maximum principle and the Bellman equation can already be found in Caratheodory's book of 1935 (Ref. 1a), the first even in his earlier work of 1926 which is given in Ref. 2. This is not a widely acknowledged fact. The present tutorial paper traces Caratheodory's approach to the calculus of variations, once called the "royal road in the calculus of variations," and shows that famous results in optimal control theory, including the maximum principle and the Bellman equation, are consequences of Caratheodory's earlier results.

Local Optimality Conditions and Lipschitzian Solutions to the Hamilton–Jacobi Equation

We consider an optimal control problem with end-constraints formulated in terms of a differential inclusion. A sufficient condition for local optimality of a trajectory is given, involving a Lipschitzian function $\phi $ which is a generalized solution to the Hamilton–Jacobi equation. It is shown that the weakest hypothesis under which the condition is also necessary is that the problem be locally calm. It is further proved that local calmness is implied by strong normality. We thereby establish that the Caratheodory approach, modified to permit Lipschitzian functions $\phi $, is applicable in principle when the first order optimality conditions yield nontrivial information.

On the Hamilton-Jacobi Equation of Differential Games

The present article considers the Hamilton-Jacobi partial differential equation of some simple differential games. This equation is developed by using Caratheodory's approach to the synthesis problem of optimal control.

Caratheodory-Hamilton-Jacobi theory in optimal control

In this paper we have presented the Caratheodory approach to the calculus of Variations as modified to suit optimal control problems. This method is by determining and solving a problem equivalent to the original problem, the Hamilton-Jacobi partial differential equation being a key step in the determination of the new problem. We have shown the relationship of the Pontryagin Maximum Principle to this method, as well as the analogs of the Weierstrass Excess Function and the classical sufficient conditions. We have also discussed the solution of the Hamilton-Jacobi partial differential equation and have illustrated the whole method with an example problem. The ideas presented in this paper are useful in the solution of optimal control problems, but their greatest value lies in a clearer understanding of the concepts underlying these problems.