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

Argon does not affect cerebral circulation or metabolism in male humans

Objective: Accumulating data have recently underlined argońs neuroprotective potential. However, to the best of our knowledge, no data are available on the cerebrovascular effects of argon (Ar) in humans. We hypothesized that argon inhalation does not affect mean blood flow velocity of the middle cerebral artery (Vmca), cerebral flow index (FI), zero flow pressure (ZFP), effective cerebral perfusion pressure (CPPe), resistance area product (RAP) and the arterio-jugular venous content differences of oxygen (AJVDO2), glucose (AJVDG), and lactate (AJVDL) in anesthetized patients. Materials and methods: In a secondary analysis of an earlier controlled cross-over trial we compared parameters of the cerebral circulation under 15 minutes exposure to 70%Ar/30%O2versus 70%N2/30%O2in 29 male patients under fentanyl-midazolam anaesthesia before coronary surgery. Vmca was measured by transcranial Doppler sonography. ZFP and RAP were estimated by linear regression analysis of pressure-flow velocity relationships of the middle cerebral artery. CPPe was calculated as the difference between mean arterial pressure and ZFP. AJVDO2, AJVDG and AJVDL were calculated as the differences in contents between arterial and jugular-venous blood of oxygen, glucose, and lactate. Statistical analysis was done by t-tests and ANOVA. Results: Mechanical ventilation with 70% Ar did not cause any significant changes in mean arterial pressure, Vmca, FI, ZFP, CPPe, RAP, AJVDO2, AJVDG, and AJVDL. Discussion: Short-term inhalation of 70% Ar does not affect global cerebral circulation or metabolism in male humans under general anaesthesia

Lazy Abstraction-Based Controller Synthesis

We present lazy abstraction-based controller synthesis (ABCS) for continuous-time nonlinear dynamical systems against reach-avoid and safety specifications. State-of-the-art multi-layered ABCS pre-computes multiple finite-state abstractions of varying granularity and applies reactive synthesis to the coarsest abstraction whenever feasible, but adaptively considers finer abstractions when necessary. Lazy ABCS improves this technique by constructing abstractions on demand. Our insight is that the abstract transition relation only needs to be locally computed for a small set of frontier states at the precision currently required by the synthesis algorithm. We show that lazy ABCS can significantly outperform previous multi-layered ABCS algorithms: on standard benchmarks, lazy ABCS is more than 4 times faster

Model order reduction approaches for infinite horizon optimal control problems via the HJB equation

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods

Grid refinement in the construction of Lyapunov functions using radial basis functions

Lyapunov functions are a main tool to determine the domain of attraction of equilibria in dynamical systems. Recently, several methods have been presented to construct a Lyapunov function for a given system. In this paper, we improve the construction method for Lyapunov functions using Radial Basis Functions. We combine this method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of data points needed to construct Lyapunov functions. Finally, we give numerical examples to illustrate our algorithms

Small intra-individual variability of the preejection period justifies the use of pulse transit time as approximation of the vascular transit

Background Vascular transit time (VTT) is the propagation time of a pulse wave through an artery; it is a measure for arterial stiffness. Because reliable non-invasive VTT measurements are difficult, as an alternative we measure pulse transit time (PTT). PTT is defined as the time between the R-wave on electrocardiogram and arrival of the resulting pulse wave in a distal location measured with photoplethysmography (PPG). The time between electrical activation of the ventricles and the resulting pulse wave after opening of the aortic valve is called the pre-ejection period (PEP), a component of PTT. The aim of this study was to estimate the variability of PEP at rest, to establish how accurate PTT is as approximation of VTT. Methods PTT was measured and PEP was assessed with echocardiography (gold standard) in three groups of 20 volunteers: 1) a control group without cardiovascular disease aged 50 years, and 3) a group with cardiovascular risk factors, defined as arterial hypertension, dyslipidemia, kidney failure and diabetes mellitus. Results Per group, the mean PEP was: 1) 58.5 ± 13.0 ms, 2) 52.4 ± 11.9 ms, and 3) 57.6 ± 11.6 ms. However, per individual the standard deviation was much smaller, i.e. 1) 2.0-5.9 ms, 2) 2.8-5.1 ms, and 3) 1.6-12.0 ms, respectively. There was no significant difference in the mean PEP of the 3 groups (p = 0.236). Conclusion In conclusion, the intra-individual variability of PEP is small. A change in PT

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov function for nonlinear systems with asymptotically stable equilibria. In it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium’s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C² and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper we have shown these results for planar systems, in this paper we cover general n-dimensional systems