How does the brain control the respiratory system?

Hint: The systema respiratorium may be a biological system consisting of specific organs and structures used for gas exchange in animals and plants. The anatomy and physiology that make this happen varies greatly, counting on the dimensions of the organism, the environment in which they live and their evolutionary history.

Complete answer:
The motor area within the cerebral mantle of the brain controls voluntary respiration (the ascending respiratory pathway).
Voluntary respiration could also be overridden by aspects of involuntary respiration, like chemoreceptor stimulus, and hypothalamus stress response.
The phrenic nerves, vagus nerves, and posterior thoracic nerves are the main nerves involved in respiration.
Voluntary respiration is required to perform higher functions, like voice control.
The primary motor area is the neural center for voluntary respiratory control. More broadly, the motor area is liable for initiating any voluntary muscular movement.
The respiratory centre within the medulla and pons of the brainstem controls the speed and depth of respiration, (the respiratory rhythm), through various inputs. These include signals from the peripheral chemoreceptors and central chemoreceptors; from the vagus and nervus glossopharyngeus carrying input from the pulmonary stretch receptors, and other mechanoreceptors within the lungs, also signals from the cerebral mantle and hypothalamus, Medulla.
Ventral respiratory group (includes the pre-Bötzinger complex). The ventral respiratory group controls voluntary forced exhalation and acts to extend the force of inhalation. Regulates rhythm of Inhalation and Exhalation.
Dorsal Respiratory group (solitary nucleus). The dorsal respiratory group controls mostly movements of inhalation and their timing.

Note:
In most fish, and a variety of other aquatic animals (both vertebrates and invertebrates) the systema respiratorium consists of gills, which are either partially or completely external organs, bathed within the watery environment.