Ukrainian presidential advisor Oleksiy Arestovych said Russian forces were trying to lay siege to Kyiv and the northeastern city Kharkiv, Ukraine's second-largest.
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In strategic Kharkiv, an eastern city with a population of about 1.5 million, videos posted online showed explosions hitting the region's Soviet-era administrative building and residential areas.
Russian troops fired artillery at Kyiv, Kharkiv and the southern port city of Mariupol overnight while the Ukrainian side shot down Russian military planes around the capital, Arestovych said in a televised briefing.
The mayor of Mariupol said on Tuesday morning that the southern port city was under constant shelling which had killed civilians and damaged infrastructure, as Russia started day six of its invasion.
"We have had residential quarters shelled for five days. They are pounding us with artillery, they are shelling us with GRADS, they are hitting us with air forces," Vadym Boichenko said in a live broadcast on Ukrainian TV.
"We have civilian infrastructure damaged - schools, houses. There are many injured. There are women, children killed," he said.
Kharkiv, near the Russian border, is another key target. One after the other, explosions burst through a residential area of the city in one video verified by AP. In the background, a man pleaded with a woman to leave, and a woman cried.
Arestovych described the current situation as under control.
An adviser to Ukraine President Volodymyr Zelenskyy said on Tuesday Russia was deliberately shelling cities, including residential areas and civilian infrastructure, to spread panic among Ukrainians.
"The veil has come down. Russia is actively shelling city centers, launching direct missile and artillery strikes on residential areas and administration sites," said Ukrainian presidential adviser, Mykhailo Podolyak.
"Russia's goal is clear – mass panic, civilian casualties and damaged infrastructure. Ukraine is fighting honorably," Podolyak said.
Meanwhile, governor of Ukraine's central bank Kyrylo Shevchenko said in a statement Tuesday that Ukraine's banking system was running smoothly and banks were replenishing cash machines where possible.
Online banking was operating as usual, allowing non-cash transfers including to the army, he said.
NATO's Chief Jens Stoltenberg called on Russia on Tuesday to end the war in Ukraine and withdraw all its forces, adding the alliance would not send troops or combat jets to support Kyiv as it does not want to become part of the conflict.
"The Russian assault is totally unacceptable and it is enabled by Belarus," Stoltenberg said on Tuesday after meeting Polish president Andrzej Duda.
"NATO is a defensive alliance, we do not seek conflict with Russia. Russia must immediately stop the war, pull all its forces from Ukraine and engage in good faith in diplomatic efforts," he added.
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