Input-to-state stability of infinite-dimensional control systems

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs the existence of an ISS-Lyapunov function implies the input-to-state stability of a system. Then for the case of systems described by abstract equations in Banach spaces we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system which linear approximation is ISS. In order to study interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations.Comment: 33 page

Moderate hyperventilation during intravenous anesthesia increases net cerebral lactate efflux

BACKGROUND:: Hyperventilation is known to decrease cerebral blood flow (CBF) and to impair cerebral metabolism, but the threshold in patients undergoing intravenous anesthesia is unknown. The authors hypothesized that reduced CBF associated with moderate hyperventilation might impair cerebral aerobic metabolism in patients undergoing intravenous anesthesia. METHODS:: Thirty male patients scheduled for coronary surgery were included in a prospective, controlled crossover trial. Measurements were performed under fentanyl-midazolam anesthesia in a randomized sequence aiming at partial pressures of carbon dioxide of 30 and 50 mmHg. Endpoints were CBF, blood flow velocity in the middle cerebral artery, and cerebral metabolic rates for oxygen, glucose, and lactate. Global CBF was measured using a modified Kety-Schmidt technique with argon as inert gas tracer. CBF velocity of the middle cerebral artery was recorded by transcranial Doppler sonography. Data were presented as mean (SD). Two-sided paired t tests and one-way ANOVA for repeated measures were used for statistical analysis. RESULTS:: Moderate hyperventilation significantly decreased CBF by 60%, blood flow velocity by 41%, cerebral oxygen delivery by 58%, and partial pressure of oxygen of the jugular venous bulb by 45%. Cerebral metabolic rates for oxygen and glucose remained unchanged; however, net cerebral lactate efflux significantly increased from -0.38 (2.18) to -2.41(2.43) μmol min 100 g. CONCLUSIONS:: Moderate hyperventilation, when compared with moderate hypoventilation, in patients with cardiovascular disease undergoing intravenous anesthesia increased net cerebral lactate efflux and markedly reduced CBF and partial pressure of oxygen of the jugular venous bulb, suggesting partial impairment of cerebral aerobic metabolism at clinically relevant levels of hypocapnia. Copyrigh

Kant's Threefold Synthesis On a Moderately Conceptualist Interpretation

In this chapter I advance a moderately conceptualist interpretation of Kant’s account of the threefold synthesis in the A-Deduction. Often the first version of TD, the A-Deduction, is thought to be less conceptualist than the later B-version from 1787 (e.g. Heidegger 1991, 1995). Certainly, it seems that in the B-Deduction Kant puts more emphasis on the role of the understanding in determining the manifold of representations in intuition than he does in the A-Deduction. It also appears that in the A-Deduction the seemingly pre-conceptual aspects of a priori synthesis, namely those of the synthesis of apprehension and the imagination, are more prominently featured than in the B-Deduction. And the fact that in the A-Deduction judgement does not appear to play any significant role reinforces the view that the A-Deduction is less strongly conceptualist. I believe that Kant is a conceptualist also in the A-Deduction (as much as in the B-Deduction) in the sense that all syntheses, which are expounded in the second section of the A-Deduction, must be seen as involving the categories or the understanding as the seat of the categories. However, despite some apparent strong modal claims regarding apperception in the A-Deduction, I argue that Kant is a moderate conceptualist in the sense that he allows for the real possibility that some representations are apprehended that are not subsumed or subsumable under the categories, or determined or determinable by the understanding as the seat of the categories. Not all representations must be synthesised and hence be conceptualised (by means of the categories), nor are all representations necessarily conceptualisable (by means of the categories). Often it is argued that the application of the categories must be seen as separate from or prior to conceptualisation (that is, employment of concepts in a judgement), so that the categories must be considered to apply to representations at least to the extent that the productive imagination or recognitive synthesis is involved, even if no empirical concepts are applied in an actual judgement. But it is difficult to see how categories can apply outside the context of an actual judgement in which ipso facto empirical concepts are employed, because, after all, categories are nothing but logical functions of judgement (e.g. B143). More in particular, I shall argue for the claims that (1) appearances to the contrary, all three levels of syntheses in the A-Deduction, including the synthesis of recognition, are interdependent and are not to be seen as operating singly or independently of each other, and hence of the categories; (2) ‘mere’ apprehension, or ‘mere’ intuition, is not dependent on the understanding and the application or possible application of the categories; and that (3) ‘mere’ apprehension does not even invoke a priori synthesis of apprehension and hence is as such fully lawless in terms of Kantian a priori laws. In this context, I also address Kant’s argument in the A-Deduction about the role of the imagination in the production of spatial objects and explain his apt use of the example of cinnabar to show that the kind of association that is at issue here concerns the possibility of knowledge, not the possibility of mere association, as is often assumed

Getting Acquainted with Kant

My question here concerns whether Kant claims that experience has nonconceptual content, or whether, on his view, experience is essentially conceptual. However there is a sense in which this debate concerning the content of intuition is ill-conceived. Part of this has to do with the terms in which the debate is set, and part to do with confusion over the connection between Kant’s own views and contemporary concerns in epistemology and the philosophy of mind. However, I think much of the substance of the debate concerning Kant’s views on the content of experience can be salvaged by reframing it in terms of a debate about the dependence relations, if any, that exist between different cognitive capacities. Below, in Section 2, I clarify the notion of ‘content’ I take to be at stake in the interpretive debate. Section 3 presents reasons for thinking that intuition cannot have content in the relevant sense. I then argue, in Section 4, that the debate be reframed in terms of dependence. We should distinguish between Intellectualism, according to which all objective representation (understood in a particular way) depends on acts of synthesis by the intellect, and Sensibilism, according to which at least some forms of objective representation are independent of any such acts (or the capacity for such acts). Finally, in Section 5, I further elucidate the cognitive role of intuition. I articulate a challenge which Kant understands alethic modal considerations to present for achieving cognition, and argue that a version of Sensibilism that construes intuition as a form of acquaintance is better positioned to answer this challenge than Intellectualism

Problems of Kantian Nonconceptualism and the Transcendental Deduction

In this paper, I discuss the debate on Kant and nonconceptual content. Inspired by Kant’s account of the intimate relation between intuition and concepts, McDowell (1996) has forcefully argued that the relation between sensible content and concepts is such that sensible content does not severally contribute to cognition but always only in conjunction with concepts. This view is known as conceptualism. Recently, Kantians Robert Hanna and Lucy Allais, among others, have brought against this view the charge that it neglects the possibility of the existence of essentially nonconceptual content that is not conceptualised or subject to conceptualisation. Their critique of McDowell amounts to nonconceptualism. However, both views, conceptualist and nonconceptualist, share the assumption that intuition is synthesised content in Kant’s sense. My interest is not in the validity of the philosophical positions of conceptualism or nonconceptualism per se. I am particularly interested in the extent to which the views that McDowell and Hanna and Allais respectively advance are true to Kant, or can validly be seen as Kantian positions. I argue that although McDowell is right that intuition is only epistemically relevant in conjunction with concepts, Hanna and Allais are right with regard to the existence of essentially nonconceptual content (intuitions) independently of the functions of the understanding, but that they are wrong with regard to non-conceptualised intuition being synthesised content in Kant’s sense. Kantian conceptualists (Bowman 2011; Griffith 2012; Gomes 2014) have responded to the recent nonconceptualist offensive, with reference to A89ff./B122ff. (§13)—which, confusingly, the nonconceptualists also cite as evidence for their contrary reading—by arguing that the nonconceptualist view conflicts with the central goal of the Transcendental Deduction, namely, to argue that all intuitions are subject to the categories. I contend that the conceptualist reading of A89ff./B122ff. is unfounded, but also that the nonconceptualists are wrong to believe that intuitions as such refer strictly to objects independently of the functions of the understanding, and that they are mistaken about the relation between figurative synthesis and intellectual synthesis. I argue that Kant is a conceptualist, albeit not in the sense that standard conceptualists assume. Perceptual knowledge is always judgemental, though without this resulting in the standard conceptualist claim that, necessarily, all intuitions or all perceptions per se stand under the categories (strong conceptualism). I endorse the nonconceptualist view that, for Kant, perception per se, i.e. any mere or ‘blind’ intuition of objects (i.e. objects as indeterminate appearances) short of perceptual knowledge, does not necessarily stand under the categories. Perception is not yet perceptual knowledge. In this context, I point out the common failure in the literature on the Transcendental Deduction, both of the conceptualist and nonconceptualist stripe, to take account of the modal nature of Kant’s argument for the relation between intuition and concept insofar as cognition should arise from it

Review on computational methods for Lyapunov functions

Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function